>You should just have a Martian Internet and then sync up the two every few hours or something.

the "internet" is just a network of computers. "sync up the two" would involve replicating the servers and coming up with some sort of consensus protocol. Needless to say, that's non-trivial to implement and needs to be done for each site individually. If there was a martian colony it might be economically feasible for some sites to do this, but if it's just a few astronauts the most feasible way of implementing internet browsing would be something like archive.today, where you send a url to some server on earth, the server runs a headless browser and takes a screenshot, and sends the captured contents back to mars/spacecraft.

The question is whether it will be like Usenet with nntp or a BBS with xmodem

It would have a 30 minute ping for rendering, but all the round trips required for loading the page would be a few orders of magnitude shorter.

Probably not possible for the first several years, with limited bandwidth. Maybe they could use sneakernet for periodic updates.