keeping awake - drink a Dome Olio coffee or a local brew which are strong and tasty (local ones r called kopi O for black and kopi for added sweetened condense milk).
Hotel - even though u arrive so early and check in time is at 3 pm, u can still try to check in at the hotel (sometimes there are rooms available and they do check u in without the need to wait till check in time) and if there is nothing available, u can leave your bags with them, ask them if u can use any of their facilities (i had often times been offered lounge use or gym and pool etc). There has never been once that I was refused. Grab something to drink, read a daily and see what is being offered for programs/activities in the city to plan your next few days in town. U can also make a trip to Little India or Chinatown for activities (at the wet market/hawker centers, locals are already going on with their daily activities of grocery shopping, having breakfast, doing tai chi or enjoying their birds in their cages or playing chinese checkers/chess etc).
Dont sleep in the morning or u wont be able to sleep when it is night time in SIN.
As for shopping, if u buy the known brands (like braun), there is no worries about it working in UK and in town, the prices are usually fixed but there is always some kind of sales or promotion going on so u might be in luck for a good price. For haggling, those will be in malls like Far East, Lucky Plaza or Mustafa, the bigger, nicer malls have pretty much a fixed price. U can also try the local stores (eg. OG or Metro) for better prices (these are the ones that seem to have some kind of sale throughout the year !).