I will be setting up a cellphone store in new york city near the West
4th train station. I would like to offer mailbox services (customer
pays monthly fee to receive mail at my address) to get additional
revenue for my store. How would I go about doing this (materials
needed, permits, processes, etc.)?

Just some background information: the store has ample space, the main
room has free space around 300 sq ft, and there are two backrooms (one
is a makeshift office of about 200 sq ft, the other is currently empty
but slightly smaller).

Also, what would be the most effective way to market and price this
service (flyers, online, etc.)?

I would suggest contacting the postmaster at your local post office. 
He/She will be able to advise you on any postal regulations and
permits you will need.