Don't listen to some of these people. The internet is meant to be a place where people get helped and its just a shame that some don't offer it.
Back to your question. The picture is a unclear but as long as your rear axle dropouts are not vertical (which those don't appear to be) you should be fine. The spacing of the rear axle might not be ideal for fixed wheelsets, I'm not sure, but its nothing a little cold setting can't solve. Just be careful when doing so as that frame looks a bit old and may be brittle.
You're going to want to replace the one piece crank (if its a one piece) for something else which means you'll need a new bottom bracket as well as a crank (chainring included). Also new wheelset, tires, tubes, cog, lockring and chain at a minimum.
You're already looking to drop at a minimum of 300 or so (cheaper if you can source used parts). At that price, you're better off buying a new bike from bikes direct that is