Moving back to San Francisco this summer after 10 long years. Got a couple questions about apartment hunting.

I am writing with a few questions concerning finding an apartment in San Francisco. I currently live in Flagstaff, AZ and am planning on moving to SF this July. I am an Air Force veteran who is currently back in school full time. I am transferring to SFSU next fall and want to get a jump on the big home hunt.

My first question involves the fact that I will be a full time student on the Post 9/11 GI Bill. I don't know if you are familiar with this, but this new GI Bill will be providing me with about $2700 a month for living expenses when I make the move to SF. Because of this I will not be seeking employment; at least not right away. Will being unemployed and receiving all of my income from the GI Bill be an obstacle for me? I look on Craigslist a lot and most places require some proof of employment, understandably. I won't actually be employed but I will be receiving almost $3k a month from the Veteran's Administration for living expenses while enrolled in school.

My second question concerns price range. I'd like to find a place in the city proper for about $1300-$1500. 1 or 2 bedroom. I understand the 2 bedroom may be pushing it, but I might get lucky. Is this price range realistic?

And finally, in May would I be able to set up some times to see some apartments that meet my criteria? It would be towards the end of May, and I will be moving to SF the second week of July. My hope is to put down a deposit and secure a place for me to live during my visit in May. That way when I move in July I'll be able to pull up and unload my U-Haul and have a home waiting for me.

Thank you in advance for taking the time to read over this!