TST: add test_ffill_with_string_column
- [x] closes #40250 
- [x] tests added / passed
- [ ] Ensure all linting tests pass, see [here](https://pandas.pydata.org/pandas-docs/dev/development/contributing.html#code-standards) for how to run them
- [ ] whatsnew entry

#40250 is an issue which seems to have been fixed in master already through #39446. This PR adds a test for this issue.

Hey @mzeitlin11, no reason in particular - just thought I'd wait for the CI checks to pass before making it ready for review.

It does seem that one of the CI checks is failing (CI / Web and docs), but I have no reason to suspect that the failure has to do with the changes in this PR.

Also: it seems that the whatsnew entry may not be needed since this is just testing an already-fixed bug. Let me know if that's the case.

I have marked as ready to review! Thank you > Hey @mzeitlin11, no reason in particular - just thought I'd wait for the CI checks to pass before making it ready for review.
> 
> It does seem that one of the CI checks is failing (CI / Web and docs), but I have no reason to suspect that the failure has to do with the changes in this PR.
> 
> Also: it seems that the whatsnew entry may not be needed since this is just testing an already-fixed bug. Let me know if that's the case.
> 
> I have marked as ready to review! Thank you

Yep, whatsnew is not necessary. The CI / Web and docs failure is unrelated and should be fixed by #40542 thanks @shiv-io 