Some plastic is okay, most is not....they usually tear fins very, very easily. Silk or live is MUCH better...
Also, I advise upgrading to at least a 2.5 gallon. Due to the fact that fish are constantly excreting ammonia through their gills(their form of urine, if you will)along with the ammonia created from their poo, a 1/2 gallon will need water changes every day to prevent the ammonia from building up to deadly levels.
Furthermore, bettas are, in fact, tropical fish and require a stable temp of 78-80 degrees F. Hes probably 'bored' because hes cold. Unless you keep your house that warm constantly, or you live in an area with a steady temp around there year round then you'll need a heater...and that requires a 2.5+ gallon tank.
It doesn't have to be expensive...a good option are Kritter Keepers from the reptile section, medium-large size. They're light, easy to carry for water changes, have a nice, secure lid, and can be safely heated with any submersible, adjustable, 25 watt heater :)
Hopefully this helps, and congrads on your new friend!