Is it possible to remotely engage in a side project with a postdoc in order to gain experience?
You end your post with multiple questions. It is usually better to try to write one focussed post per question on these sites. However, I will focus on two questions that I feel are key here:

Would it be possible to do this project remotely?
How should you contact the postdoc?



Would it be possible to do this project remotely?


You can always ask to do a side project, research is about collaboration. Remote collaboration should be fine, given that that is how a lot of companies and universities are functioning now due to the pandemic. Zoom/Teams/Slack or something similar should work fine, use whatever the postdoc is using. Do go to your advisor first and see if they agree that this fits in your PhD or if you should do this on your own time.


How should you contact the postdoc?


An unprompted message might go unnoticed. Ask your professor if they by any chance know the postdoc you want to collaborate with. If they do, ask if they can introduce you and already vouch for your work. If they don't, look for another possible connection to do this. This will give you the highest probability of success. If you end up not being able to find somebody to help you, send them a message directly. Keep your message brief, direct and describe your background and how you'd like to help (depending on the topic you could help with for example data cleaning, analysis or visualization, literature reviews, etc.) and ask them if they have something for you, now or in the future.