BUG: DataFrame.update doesn't preserve dtypes
Preserve dtype when updating from dataframe whose NA values don't affect original.

I don't know the best place to put the test case in the tests/frame or the tests/dtype directory.

- [x] closes #55509
- [x] [Tests added and passed](https://pandas.pydata.org/pandas-docs/dev/development/contributing_codebase.html#writing-tests) if fixing a bug or adding a new feature
- [x] All [code checks passed](https://pandas.pydata.org/pandas-docs/dev/development/contributing_codebase.html#pre-commit).
- [ ] Added [type annotations](https://pandas.pydata.org/pandas-docs/dev/development/contributing_codebase.html#type-hints) to new arguments/methods/functions.
- [ ] Added an entry in the latest `doc/source/whatsnew/vX.X.X.rst` file if fixing a bug or adding a new feature.

I found pandas/tests/frame/methods/test_update.py and will update test case there. This pull request is stale because it has been open for thirty days with no activity. Please [update](https://pandas.pydata.org/pandas-docs/stable/development/contributing.html#updating-your-pull-request) and respond to this comment if you're still interested in working on this. My changes obsoleted by https://github.com/pandas-dev/pandas/pull/55634