A 10g is better. I do understand about not having the money for this kind of thing. Really though, the only anemone that will not outgrow this size tank is a flower. Even bubble tips get large, so do condys, and carpets get huge.....
Another thought is that all anemones require good lighting. For a 10g which is a shallow tank, you could get by with PC's (power compacts) for a flower anemone. PC's are rather expensive, more so than regular fluorescents but they will be worth it in the long run.
You also need to realise that all anemones sting. All of them. Some are fish eaters. In a small tank, this can be a real problem. I'm not saying that you should never get an anemone, just maybe wait awhile until you get some experience with saltwater set ups, and can provide the right home for one.
Things you need for a 10g include lighting (we'll talk about this later), heater, thermometer, hydrometer, 2 powerheads, filter (which many consider optional in a small tank. Instead, many believe that weekly water changes and the live rock in the tank will replace one. Frankly I don't go with that. I like having a filter on the tank, not only does it provide a place to run carbon/floss but it also provides another current), live rock anywhere from 5 to 15 lbs of it, sand, test kits.
Cycling can take anywhere from 2 weeks to 2 months....depends.
I would use the fishless method to cycle the tank. Simply drop a piece of raw fish or shrimp into the tank and let it decay. This will provide the ammonia as it dissolves. Many people use a live fish like a damsel to cycle their tank with. The fishless method not only saves you money on buying the fish, it saves the fish from the stress of cycling (and maybe it's life as many die during this), and it saves you the frustration of catching a fish when you no longer want the damsel in your tank.
Lighting....even without an anemone, I would still go with the PC's if you can. This way you could keep just about anything in there without worry and without the hassle of upgrading later on.
Things to consider....
Even with a filter on the tank, I would still do weekly water changes. Things can go wrong quickly in such a small volumne of water.
Be very careful not to overstock your fish. In a 10g I would not keep any more than 2 small fish, and I do mean at adult size they are small.
Start out with hardy inverts like featherdusters, mushroom corals, button polyps.......
It might be a good idea to go with some macro algae too, like feather calupera. They will help to take up excess nutrients.
Read, read, read!
HTH