Is it possible to patch the game on my friends PS3 and then play the game patched on my PS3?

what?

you both have to download it after signing into PSN. unless you guys are sharing a system, you both have to download it. it will download automatically when you start the game, but you need to be signed in.

I don't have internet where my PS3 is...

sorry, dude, but unless you get your system an internet connection, your out of luck. are you able to take your system to your friends house? if you can get anywhere with a wireless connection and a TV your set.

another thing you can do; when I didn't have Internet and I first got my system, I was pretty chills with my old neighbour, who has a PS3 as well. he let me use his Internet to get patches and play online sometimes, and at the end of the month I would toss him 10-15 bucks. depending how much I used. then for a little while someone set up a WLAN with no security settings, so I just used that. if you scan every now and then you may get lucky.