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.

thanks @lithomas1  Owee, I'm MrMeeseeks, Look at me.

There seem to be a conflict, please backport manually. Here are approximate instructions:

1. Checkout backport branch and update it.

```
$ git checkout 1.3.x
$ git pull
```

2. Cherry pick the first parent branch of the this PR on top of the older branch:
```
$ git cherry-pick -m1 f61347bfb48862b8657c59f63bce6d9c16119e44
```

3. You will likely have some merge/cherry-pick conflict here, fix them and commit:

```
$ git commit -am 'Backport PR #42538: CI: Run some checks on push on 1.3.x'
```

4. Push to a named branch :

```
git push YOURFORK 1.3.x:auto-backport-of-pr-42538-on-1.3.x
```

5. Create a PR against branch 1.3.x, I would have named this PR:

> "Backport PR #42538 on branch 1.3.x"

And apply the correct labels and milestones.

Congratulation you did some good work ! Hopefully your backport PR will be tested by the continuous integration and merged soon!

If these instruction are inaccurate, feel free to [suggest an improvement](https://github.com/MeeseeksBox/MeeseeksDev).
                 can you push the backport PR as well @lithomas1 ty