First, if you can swing it I'd recommend getting a server with a static IP. You can get can get them as cheap as a few bucks a year from https://lowendtalk.com/categories/offers ofc with that they may have more downtime than 5/mo boxes.

Anyway, if you don't wanna pay you can use something like duckdns for dynamic DNS and get a free domain that automatically points to your new IP whenever it changes. Works well if you're hosting from home. :)