how do i watch the activity of one user in my private repository?

I have a private repository. Need getting user activity (pull requests, comments filtered by user). 

GET /repos/:owner/:repo/pulls -  can I filtered this query by user login (id or email) ?

For that particular endpoint, you can’t instruct GitHub to filter it by username or email for you, no. You’ll need to get the results and then filter them yourself. You could, however, use the search endpoint and create a search that finds only the activity by that user and use that.

I hope that helps! Let us know if you have any other questions.