diff --git a/.github/workflows/r.yml b/.github/workflows/r.yml
index 80dd30335f8..01cd2b1250d 100644
--- a/.github/workflows/r.yml
+++ b/.github/workflows/r.yml
@@ -64,13 +64,11 @@ env:
 jobs:
   ubuntu-minimum-cpp-version:
     name: Check minimum supported Arrow C++ Version (${{ matrix.cpp_version }})
-    # We don't provide Apache Arrow C++ 15.0.2 deb packages for Ubuntu 24.04.
-    # So we use ubuntu-22.04 here.
     runs-on: ubuntu-22.04
     strategy:
       matrix:
         include:
-          - cpp_version: "15.0.2"
+          - cpp_version: "19.0.1"
     steps:
     - name: Checkout Arrow
       uses: actions/checkout@3df4ab11eba7bda6032a0b82a6bb43b11571feac # v4.0.0
diff --git a/dev/tasks/r/github.linux.arrow.version.back.compat.yml b/dev/tasks/r/github.linux.arrow.version.back.compat.yml
index fae8630d132..ecbea54ce49 100644
--- a/dev/tasks/r/github.linux.arrow.version.back.compat.yml
+++ b/dev/tasks/r/github.linux.arrow.version.back.compat.yml
@@ -66,19 +66,7 @@ jobs:
   read-files:
     name: "Read files with Arrow {{ '${{ matrix.config.old_arrow_version }}' }}"
     needs: [write-files]
-    # condition && true-case || false-case
-    # ==
-    # condition ? true-case : false-case
-    #
-    # We use Ubuntu 20.04 for 3.0.0 or earlier because 3.0.0 or
-    # earlier can't be built on Ubuntu 22.04. We will drop 3.0.0 or
-    # earlier when Ubuntu 20.04 is unavailable on GitHub Actions.
-    runs-on: >-
-      {{ "${{ (startsWith(matrix.config.old_arrow_version, '3.') ||
-               startsWith(matrix.config.old_arrow_version, '2.') ||
-               startsWith(matrix.config.old_arrow_version, '1.')) &&
-                 'ubuntu-20.04' ||
-                 'ubuntu-22.04' }}" }}
+    runs-on: 'ubuntu-22.04'
     strategy:
       fail-fast: false
       matrix:
@@ -100,10 +88,7 @@ jobs:
         - { old_arrow_version: '7.0.0', r: '4.1' }
         - { old_arrow_version: '6.0.1', r: '4.1' }
         - { old_arrow_version: '5.0.0', r: '4.1' }
-        - { old_arrow_version: '4.0.0', r: '4.0' }
-        - { old_arrow_version: '3.0.0', r: '4.0' }
-        - { old_arrow_version: '2.0.0', r: '4.0' }
-        - { old_arrow_version: '1.0.1', r: '4.0' }
+        - { old_arrow_version: '4.0.0', r: '4.1' }
     env:
       ARROW_R_DEV: "TRUE"
       OLD_ARROW_VERSION: {{ '${{ matrix.config.old_arrow_version }}' }}
@@ -118,11 +103,6 @@ jobs:
             # for 6.0.0 or later.
             rspm="https://packagemanager.rstudio.com/cran/__linux__/jammy/latest"
             echo "RSPM=${rspm}" >> $GITHUB_ENV
-          elif [ ${old_arrow_version_major} -le 3 ]; then
-            # We use Ubuntu 20.04 for 3.0.0 or earlier because 3.0.0 or earlier
-            # can't be built on Ubuntu 22.04.
-            rspm="https://packagemanager.rstudio.com/cran/__linux__/focal/latest"
-            echo "RSPM=${rspm}" >> $GITHUB_ENV
           fi
       - uses: r-lib/actions/setup-r@v2
         with:
@@ -135,8 +115,7 @@ jobs:
       - name: Setup our testing directory, copy only the tests to it.
         run: |
           mkdir -p extra-tests/files
-          cp arrow/r/extra-tests/helper*.R extra-tests/
-          cp arrow/r/extra-tests/test-*.R extra-tests/
+          cp arrow/r/extra-tests/*.R extra-tests/
       - name: Download artifacts
         uses: actions/download-artifact@v4
         with:
diff --git a/r/extra-tests/test-read-files.R b/r/extra-tests/test-read-files.R
index 2f5db38e520..4d4ecf85faf 100644
--- a/r/extra-tests/test-read-files.R
+++ b/r/extra-tests/test-read-files.R
@@ -18,7 +18,21 @@
 library(arrow)
 library(testthat)
 
-source("tests/testthat/helper-skip.R")
+# These are similar to functions in tests/testthat/helper-skip.R but we duplicate them
+# here since very old versions don't have arrow_info() with exactly the same shape
+if_version <- function(version, op = `==`) {
+  op(packageVersion("arrow"), version)
+}
+
+if_version_less_than <- function(version) {
+  if_version(version, op = `<`)
+}
+
+skip_if_version_less_than <- function(version, msg) {
+  if (if_version(version, `<`)) {
+    skip(msg)
+  }
+}
 
 pq_file <- "files/ex_data.parquet"
 
@@ -32,7 +46,7 @@ test_that("Can read the file (parquet)", {
 
 ### Parquet
 test_that("Can see the metadata (parquet)", {
-  skip_if_arrow_version_less_than("2.0.0", "Version 1.0.1 can't read new version metadata.")
+  skip_if_version_less_than("2.0.0", "Version 1.0.1 can't read new version metadata.")
 
   df <- read_parquet(pq_file)
   expect_s3_class(df, "tbl")
@@ -76,7 +90,7 @@ for (comp in c("lz4", "uncompressed", "zstd")) {
   })
 
   test_that(paste0("Can see the metadata (feather ", comp, ")"), {
-    skip_if_arrow_version_less_than("2.0.0", "Version 1.0.1 can't read new version metadata.")
+    skip_if_version_less_than("2.0.0", "Version 1.0.1 can't read new version metadata.")
 
     df <- read_feather(feather_file)
     expect_s3_class(df, "tbl")
@@ -134,7 +148,7 @@ test_that("Can read the file (parquet)", {
 })
 
 test_that("Can see the metadata (stream)", {
-  skip_if_arrow_version_less_than("2.0.0", "Version 1.0.1 can't read new version metadata.")
+  skip_if_version_less_than("2.0.0", "Version 1.0.1 can't read new version metadata.")
   df <- read_ipc_stream(stream_file)
 
   expect_s3_class(df, "tbl")
diff --git a/r/tests/testthat/helper-skip.R b/r/tests/testthat/helper-skip.R
index fdfe152d6ee..eb69fb55dc4 100644
--- a/r/tests/testthat/helper-skip.R
+++ b/r/tests/testthat/helper-skip.R
@@ -130,22 +130,6 @@ skip_on_python_older_than <- function(python_version) {
   }
 }
 
-skip_if_arrow_version_less_than <- function(version, msg) {
-  if (arrow_cpp_version() < numeric_version(version)) {
-    skip(msg)
-  }
-}
-
-skip_if_arrow_version_equals <- function(version, msg) {
-  if (arrow_cpp_version() == numeric_version(version)) {
-    skip(msg)
-  }
-}
-
-arrow_cpp_version <- function() {
-  numeric_version(gsub("-SNAPSHOT", "", arrow::arrow_info()$build_info["cpp_version"]))
-}
-
 process_is_running <- function(x) {
   if (force_tests()) {
     # Return TRUE as this is used as a condition in an if statement
diff --git a/r/tests/testthat/test-compute-arith.R b/r/tests/testthat/test-compute-arith.R
index 602e718b71c..7d77b33f262 100644
--- a/r/tests/testthat/test-compute-arith.R
+++ b/r/tests/testthat/test-compute-arith.R
@@ -235,8 +235,6 @@ test_that("Math group generics work on Array objects", {
 })
 
 test_that("hyperbolic trig functions work on Array objects", {
-  skip_if_arrow_version_less_than("18.1.0.9000", "Hyperbolic trig functions not available until version 19.")
-
   expect_equal(sinh(Array$create(c(0.6, 0.9))), Array$create(sinh(c(0.6, 0.9))))
   expect_equal(cosh(Array$create(c(0.6, 0.9))), Array$create(cosh(c(0.6, 0.9))))
   expect_equal(tanh(Array$create(c(0.6, 0.9))), Array$create(tanh(c(0.6, 0.9))))
@@ -248,7 +246,5 @@ test_that("hyperbolic trig functions work on Array objects", {
 })
 
 test_that("expm1 works on Array objects", {
-  skip_if_arrow_version_less_than("18.1.0.9000", "expm1 not available until version 19.")
-
   expect_equal(expm1(Array$create(c(0.00000001, 10))), Array$create(expm1(c(0.00000001, 10))))
 })
diff --git a/r/tests/testthat/test-dplyr-funcs-math.R b/r/tests/testthat/test-dplyr-funcs-math.R
index 07fab645e7b..e2b622cdb6e 100644
--- a/r/tests/testthat/test-dplyr-funcs-math.R
+++ b/r/tests/testthat/test-dplyr-funcs-math.R
@@ -339,8 +339,6 @@ test_that("trig functions", {
 })
 
 test_that("hyperbolic trig functions", {
-  skip_if_arrow_version_less_than("18.1.0.9000", "Hyperbolic trig functions not available until version 19.")
-
   # Note: We change df mid-test because domains differ by function
   df <- tibble(x = c(seq(from = 0, to = 1, by = 0.1), NA))
 
@@ -495,8 +493,6 @@ test_that("sqrt()", {
 })
 
 test_that("expm1()", {
-  skip_if_arrow_version_less_than("18.1.0.9000", "`expm1()` not available until version 19.")
-
   df <- tibble(x = c(1:5))
 
   compare_dplyr_binding(