bigBananas

Basically every serious mailserver package has imapsync included. I don't know what kind of hardware you're running and what you expect from the server doing the copying but running webmail, search indexes, antivirus, etc will cost you ram. There are solutions aimed at keeping backups of your mail though, I can't give you recommendations, the only thing I remember is that they don't give you a normal webmail or imap, instead you get a web-interface aimed at searching through the backup(s)

I adored my HP Veer, I bet that if I get a working one in my hands I'll blind-type flawless again in seconds, I was better on that thing than any full size keyboard. Best phone I've ever had because of the keyboard and the ease of customization when you know Javascript. Screen was a bit too small (I consider every modern phone way, way too big) and speed could've been better but wasn't bad.