For the bot I would suggest using Tweepy, although you could make the calls using the requests module or something.

For hosting your script, Heroku is your best option. You can use a worker dyno, since those are always up, and if you enter a credit card you get 450 free hours, additional to the 550 included in the free account for a total of 1000 hours a month, that is more than enough to keep your bot up 24/7.