CI: Run some checks on push on 1.3.x
- [ ] xref #39696
- [ ] 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

Not turning on CI/Checks since that one is non-trivial since it requires configuring the doc build to not upload on 1.3.x.
