Nearest airport is London City.
CityJet flies there from Dublin and while they generally - always - are more expensive than RyanAir, you don't have the trek from Stansted (Express to Liverpool Street, tube to Bank, DLR to Docklands about an hour and a half) or Luton (Train to King's Cross, two tubes to Bank, DLR probably two hours) or Gatwick (train to London Bridge, tube to Canary Wharf over an hour I would guess) to contend with. I won't even mention the Heathrow, London is not a good city to have to cross from West to East or vice versa!
Connections from City airport are by bus or by the DLR - the Docklands Light Railway - (check
Transport For London for details of all transport options, buses, ferries, trains, tubes, trams, etc).
For the O2 arena, there are several hotels in the vicinity, I would recommend getting one near Canary Wharf as there are excellent transport links from there and it is a nice area. Plus as it is a business area, hotels are less busy at the weekends. Make sure the hotel you choose is near Canary Wharf as Docklands can be quite rough in places! (Ever heard of Tower Hamlets...)
The DLR will take you within a few minutes to the Canary Wharf area. There are several hotels in the vicinity, only one I can recommend from experience is the http://www.hotels.co.uk/hotel-united-kingdom/hotel-london/marriott-west-india-quay/ (Marriott West India Quay). Stayed there once and found it perfect, the DLR stops right outside and it is within a short walk of the tube. Rooms are generously sized, the food is good and I had a wonderful view too. Plenty of shopping in the vicinity as well as there is a sizeable and rather posh shopping centre there. Oxford Street is about 40 minutes away by tube but to be honest, I hate it. It is very crowded and quite seedy in places. Covent Garden is about the same distance but much nicer for browsing. Also try and find
Seven Dials, it is a rather nice little shopping area but not so touristy as Covent Garden. Anything specific you are interested in shopping for? Or since you are nearby, pop into Greenwich, which is well worth a visit.
Oh and apply for an , definitely the cheapest way to use public transport. It is what the Luas Smart Card should be ... but isn't!!