ENH: bfill for interval indexing added GH#51503 #51503 
- [x] closes #51503
- [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).


Test is done for `backfill/bfill` methods with time intervals. `nearest` and `pad/ffill `will be added in future commits. Thanks for your pull request @hamedgibago. I noticed this is your first pull request into pandas, so I would recommend first tackling an issue labeled `good first issue`. The issue you are tackling is a bit more complex and has not been triaged by a maintainer yet  > 

I'll do `good first issue` issues. Despite this comment you mentioned and not all features added for `bfill`, because All checks have passed and green, will be this PR accepted? I am dubious that that for looping solution over each element will be performant enough for this feature > I am dubious that that for looping solution over each element will be performant enough for this feature

I've added this for type of date time check. Let me see if `self._maybe_cast_listlike_indexer` and `self._maybe_convert_i8(t)` will be removed any error happen or not. 
For the second loop of comparing indexes, I'll think another way, as I mentioned in `TODO` comment in the code Thanks for this pull request, but it appears there hasn't been much traction on this PR in a while so closing for how. Happy for you to tackle an issue that has been triaged or a `good first issue`