I never sterilised her cup, we started giving water from around 4 months due to a spell of very hot weather.
You can buy free flowing cups with a fold down spout when out and about. We have a couple of the TT ones, but Tesco have ownbrand ones too I noticed the other day for about £2 each.
Oh, and yes, from 6 months you don't need to use cooled, boiled water and can just use tap water.
Apologies for lack of capital letters - Zoe has stolen my shift key 