Congrats on the new fishy!! I can't really tell without my glasses but he looks like a twin to my Spike.
Most bettas love having a big home. They love to explore everything and swim everywhere. Make sure you provide plenty of hiding places. And if the water flow bothers him go ahead and make a baffle.
I used this process to make my baffle: http://www.bettafish.com/showthread.php?t=30139
Other users perfer using a sponge to baffle their filters. I've yet to figure that out, so currently the bottle will stay.
Changing the water in that small cup maybe a bit difficult. If you can, pour a little bit out/use your hand to block the fishy from falling out. Then get a separate cup, fill it with some tap water, add dechlorinator(i cannot spell tonight), let it sit and get the same temp as his current water/slowly pour into his cup.
We're always here to give a helping hand :)