Window AC Clearances
You should check the manual that came with the AC, it might tell you minimum clearances. My window AC unit only says:
Side louvers and the rear of the air conditioner must have clear air space to allow enough airflow through the condenser, for heat removal.
I would say that 1" does not meet that requirement, particularly if the sides of the sleeve extend out past the end of the AC.
Sizing the AC
You didn't directly ask about the difference between the window units and the thru-wall models, but I agree with @Vitality that the price jump probably has a lot to do with the size of the unit. If you're trying to cool 750 ft^2 you're going to want to get a bigger unit than the $99 budget models they sell at Walmart. According to this Energy Star chart, you'll need a unit with around 18,000 BTUs/hr of cooling. If you get a unit that's too small it will just run and run but not cool or dehumidify the apartment, wasting you money. Plus it will probably burn out the compressor faster since it's not getting a break.
How To Cool An Apartment
Besides the AC, here are things that can help you keep the apartment cool:
Get curtains and close them during the day, especially if you're not home. Preferably blackout curtains. Lots of heat enters the house as solar radiation and you can prevent it if you keep the light out. If you're on a budget, Home Depot et al sell blackout roller blinds that are pretty reasonable.
Switch your light bulbs to LEDs (or CFL, if your budget requires... but spring for the LEDs if you can, they're better and will save you money). A 100 watt incandescent light bulb puts out almost 350 BTUs/hr in heat. If you have a bunch of those on, that's going to really going to add up.
If the temperature is cool at night, fans in the window can help exhaust hot air and bring in cool air from outside. I'm not a big fan of fans just circulating air inside, like ceiling fans and tower fans, since they don't actually cool the air. But some people like to feel the breeze. Definitely shut those fans off when you leave otherwise you're just wasting electricity and heating up the room.
Use the stovetop exhaust fan when you're cooking. It will remove the hot, humid air. Same thing for exhaust vents in the bathroom during/after a shower.
Basically anything else that reduces your electrical usage will also reduce heat in the apartment. E.g. I like to put my rice cooker on the porch even if I'm not eating outside.