Sweden is a very expensive country. The Arlanda Express from the airport to the city can be around £60 return (cheaper at weekends and for couples). Need to consider these costs too as they build up quickly.
Arlanda Express (train) is quite expensive.
Flygbussarna (bus) is cheaper and stops at more places in Stockholm, and if you are lucky, your hotel might be within walking distance from a bus stop. Tickets are cheaper if you buy them in advance on the Internet.
The cheapest way is to take local bus 583 to Märsta Station and then the local SL train to Stockholm, but this takes around an hour. Buy the ticket in a shop (such as Pressbyrån) as tickets aren't sold on board the bus. If you have some ticket which allows unlimited travel in Stockholm (such as a day pass), then this ticket covers the bus and the SL train.
Trains (other than Arlanda Express) depart from Arlanda Central Station and you have to pay a station fee of 120 kr to pass through the ticket gates. The fee is sometimes included in the ticket price. If so, keep your ticket until you have passed the gates as you will otherwise have to pay the fee again. If you wish to save money, you might wish to avoid taking a train from this station. Also, note that both SL and SJ trains stop at this station, and SL and SJ use different tickets.
If you are thinking of taking a taxi, first check the price list. There should be a yellow sticker on one of the windows. Foreigners sometimes miss this and end up taking a taxi which costs 10 or 100 or 1000 times more than most other taxis.
Alternatively, take bus 801 to Uppsala instead and spend the night there. Tickets can be bought on board the bus (cards only, no cash).