Hi Sonia, and welcome to PP!
I'll try answer your questions generally, then you can ask more specific stuff as you need to:
Part-time jobs - not a great deal around at the moment, but if you're flexible you should be able to find something -best way to do this is visit potential employers once you've arrived - vast majority of hiring and firing takes place face to face over here
Banking - very easy to open accounts with any of the banks, just need your passport
General living - chilled out, laid back and generally a lot slower pace than you'll be used to!
If you're looking for somewhere to rent long term, here's a couple of sites with a very good selection:
http://www.cyprusresalesdirect.comhttp://www.the-propertypeople.comGood luck with the move!