Before I invest in the (large - the saw is 5tpi) file needed, is it possible to sharpen them?
Yes. Like all old saws this is intended to be user-sharpenable.
I'm assuming a cross cut because that's probably what I'd use it for, but maybe these are always rip?
AFAIK these and most similar saws are made for 'bucking' and other crosscutting jobs.
And certainly this pattern — the Great American — is specifically designed for cross-grain cuts. There are minor variations in the various saws identified as having this tooth pattern, as you'll see if you compare your teeth groups and gullets (in width and depth, plus taper/no taper) to other Great American saws.
Sharpening
Good news on the sharpening front is that the files originally intended to sharpen this and similar tooth patterns (which have a teardrop cross-section) are still made and still available, see here and here.1
Then see this thread on crosscutsawyer.com Although it may be worth hunting down further info that clarifies certain details of the tooth geometry and filing method I think the thread has enough information to get anyone started sharpening a saw of this type.
1 I want to also note that these files seem relatively abundant on the seconhand market so vintage files are another possibility. Even if rusty a vintage or older file may be restorable to good user condition, see previous Q&A How to restore old files? for more if interested.