Is it DIYable? Yes. No. Maybe?
You may find that you start the job, everything works out just right, and you complete the work in record time and under budget. More likely, you'll run into a bunch of unforeseen problems, take 5 times as long as you'd expect, and end up way over budget.
The best way to tackle DIY work that you are not sure about, is to have the resources available to quickly fix mistakes. In your situation, it might be a good idea to purchase a replacement window before you begin. If you damage the old window beyond repair, you'll simply have to install the new one. On the other hand, if the repair works out, you can return the new window you purchased.
It also never hurts to have somebody with knowledge in the type of work you're doing, so if you run into trouble they can offer guidance.
If you come across something you are really not comfortable with, there are always professionals available to do the work for you. You might also be surprised that they charge less than you might think (not always the case, but does happen).
First I will answer the question. Remove the window and reinstall it. You may need to adjust the lower sill or cut a little out of the header at the top to get it to set level. It is a non bearing wall so a 1/2" off the header will not matter. The sill will not have any extra material to cut out. Or you could simply resquare the window and let it run downhill like it already is.
You will not be able to trim any of the window sash to get it to fit back in without destroying the vinyl cladding, so it has to be reset.
Now for the real issue. The picture does not reveal any other symptoms but I will tell you what I see going on with the window
The wall to the left has settled for some reason, if it was originally a porch and not a living space, it may have not been structured the way a living space is needed. Either the footings where not sized properly and the footer settled down some or perhaps termites or carpenter ants got in there and caused some settling.
I would consider doing something with that, and it may not crack the window on it way back up, but when the window is corrected it will stay.
Best Answer
The answer is yes; if you use that nail-on window the way it was designed to be used it nails to the underlying wall studs. If you use a "nail-on" window like you describe, face-nailing it to the old window jambs, it will look like crap and will probably leak.
If you do not intend to purchase a window designed for your application (a block frame or "replacement style" window), an option might be for you to cut off the nail flanges so you end up with a window that has what is essentially a box/block frame. Then drill holes through the frame and screw it to the existing jambs, caulking all the way around and finishing it off with aesthetically pleasing trim of some sort. I have done this with success.
The picture above shows frame types, from left to right: