diff --git a/compose.yaml b/compose.yaml
index 6809d6994b6c..8539745e55b9 100644
--- a/compose.yaml
+++ b/compose.yaml
@@ -2044,9 +2044,14 @@ services:
       TEST_APT: 0  # would require docker-in-docker
       TEST_YUM: 0
       USE_CONDA: 1
+    # Ubuntu 24.04 needs extra tzdata-legacy, unavailable in 22.04 wheel image
     command: >
       /bin/bash -c "
-        apt update -y && apt install -y curl git gnupg tzdata wget &&
+        apt update -y &&
+        apt install -y curl git gnupg tzdata wget &&
+        if apt-cache show tzdata-legacy >/dev/null 2>&1; then
+          apt install -y tzdata-legacy;
+        fi &&
         git config --global --add safe.directory /arrow &&
         /arrow/dev/release/verify-release-candidate.sh $${VERIFY_VERSION} $${VERIFY_RC}"
 
diff --git a/dev/tasks/tasks.yml b/dev/tasks/tasks.yml
index af8b319dca3f..9d41f2ad012e 100644
--- a/dev/tasks/tasks.yml
+++ b/dev/tasks/tasks.yml
@@ -272,7 +272,6 @@ tasks:
 
 {% for distribution, version in [("conda", "latest"),
                                  ("almalinux", "10"),
-                                 ("ubuntu", "22.04"),
                                  ("ubuntu", "24.04")] %}
   {% for target in ["cpp",
                     "integration",
@@ -284,6 +283,9 @@ tasks:
     params:
       env:
         {{ distribution.upper() }}: "{{ version }}"
+        {% if distribution == "conda" %}
+        UBUNTU: "24.04"
+        {% endif %}
       target: {{ target }}
       distro: {{ distribution }}
   {% endfor %}