If your going to divide a tank you need at least a 5 gallon tank. If you got a three gallon you would only be able to have one betta (and possibly a tank mate such as a snail or shrimp).
The tank on your link is a pretty good one. They didn't list the dimensions on the link so you would just have to go to wally world and look at the box. The only problem is depending on how sturdy the shelf you want to put it on. A five gal glass tank with gravel and water (not to mention the rest of the stuff) is gonna be pretty heavy. Personally I wouldn't put anything over a 3 gal on a shelf (and thats with a low amt of gravel).. but it all depends on how sturdy the shelf is.
If you were to get the 3 gal you really still need a heater. Nothing makes betta fish thrive more than good food, clean water, and warm temps. Ideally a betta should be in water anywhere from 76-82* depending on the personality of your fish. You can get heaters really cheap at walmart as well or even on the occassional sale at petsmart.
Now as for cycling (you will still have to cycle a three gal as well)....
There are two ways to cycle depending on preference. Fish out or fish in.
Fish out requires you to add ammonia or fish food to the tank (with out fish) and test daily until you read 0ppm Ammonia, 0ppm NitrItes, and 5-20ppm NitrAtes.
Fish in (my preference) allows you to set up your tank and add your fish. What you will do is get a liquid test kit and test your water daily. Anytime you see a reading of ammonia or nitrites do a small water change (like 20-30% depending on the amt in your reading). This keeps your fish safe from being harmed by the ammonia and nitrites. You test everyday until your readings are 0 ammonia, 0 nitrites, and 5-20ppm Nitrates... these should stay the same for roughly a week.
After that you are officially cycled and don't need to test every day or do pwcs everyday.
If you get a 5 gall the perks are 2 fish (if you want to divide it but thats only 2.5 gal a fish remember), and less water changes once you are cycled. A 5 gal cycled would need 1 50-60% water change a week with gravel vacuuming.
A three gal does require 2 50% pwcs a week with gravel vacuuming during one of those changes... but if its what fits go for it. I have a 3 gallon eclipse and I love it.. enough room for my fish but still fits on my desk nicely.
The cycle process seems tedious but really its not as bad as it sounds.. you get into a routine eventually. For a 3 or 5 gallon your cycle could last anywhere from 2-6 weeks (with probably a month being average).
I just gave a quick over view of both cycles. I personally prefer the fish in cycle but some dont feel comfortable with exposing their fish to ammonia (which is poisonous to bettas) but buy doing daily small water changes that really isn't a factor. If you want more detailed info about cycling you can look in the other thread (I'll get you the link in a sec) where I went into more detail... or if you have specific questions feel free to ask :)
here's the link.. I went into a little more detail about fish in cycles on page 1 (at the bottom) http://www.bettafish.com/showthread.php?t=34239