Can you restore a project on a new computer?

Even if the session file is backed up on the cloud somewhere (presumably you know where?), that won't give you the actual working files back, because they aren't backed up at all unless you deliberately undertake to do this. Backing them up on the cloud wouldn't actually be a very clever thing to do if they were at all precious - you really ought to do that locally.

Unless it's actually the drive that's failed, then your best bet is to remove the old one from the Mac and put it into an external USB enclosure. Then you'll be able to read whatever's on it directly to your new Mac. There is a load of information available on the web about this - just do a search using the term 'convert mac internal hard drive to external' and you'll find all you need to know.