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GH-37971: [CI][Java] Don't use cache for nightly upload (#37980)

### Rationale for this change

It's too large (8.6 GB) and it's always re-created because it uses github.run_id for key. It expires other caches soon.

### What changes are included in this PR?

Remove actions/cache.

### Are these changes tested?

No.

### Are there any user-facing changes?

No.
* Closes: #37971

Authored-by: Sutou Kouhei <kou@clear-code.com>
Signed-off-by: Sutou Kouhei <kou@clear-code.com>