TST: repr of Arrow-backed object (GH #48238)
- [x] closes #48238 
- [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).
- [N/A] Added [type annotations](https://pandas.pydata.org/pandas-docs/dev/development/contributing_codebase.html#type-hints) to new arguments/methods/functions.
- [N/A] Added an entry in the latest `doc/source/whatsnew/vX.X.X.rst` file if fixing a bug or adding a new feature.

The actual issue in https://github.com/pandas-dev/pandas/issues/48238 was fixed already, only test is needed - I basically just stole the test from https://github.com/pandas-dev/pandas/issues/48238#issuecomment-1287213150. 

Also, this is my first time contributing here so sorry if I get the process wrong. Thanks!
Not sure what to do about the failing CI jobs since they don't seem to be coming from the file I touched :-(  Ah so I ran into this same issue in https://github.com/pandas-dev/pandas/pull/48258. I think there's some cached state in the testing causing unrelated tests to fail.

Would be very welcome for you to dig into why this is happening, but I would say this isn't a `good first issue` necessarily. Let me know if you'd like to continue otherwise I recommend tackling a separate issue. Okay, I don’t think I’m the best person to tackle the broken CI issue unfortunately, as I’m not super knowledgeable about advanced aspects of GHA or whatever else could be causing cache problems. Thanks for letting me know @a-reich. I'll close this pull request for now then to maybe let another user tackle this if interested. Can reopen if you are interested in tackling this issue again 