There are two smart card (IC payment card) in the Tokyo area. One, the Suica, is issued by JR East and a handful of other affiliated companies (like Tokyo Monorail), the other, Pasmo, is issued by the other Tokyo area rail/subway networks. Both cards are interchangeable. You can use them for purchases at many convenience stores, most vending machines, and most baggage lockers at train stations.
You can also add commuter passes to Suica cards. Don't do this. Last trip, my friend somehow added a pass and ended up with a months worth of travel on any JR line to any JR station between Shinjuku and Akiba. She was there for 4 days and paid for a 30 day pass.
Public transport in Japan is distance based fares. The ticket gates at a station will let you through if there is enough on the card to go one station. If you don't have enough for the station you get off at, the exit gates won't let you through and you will either need to top up the card at the nearby fare top up machine, or pay the difference at the manned booth near the gate line.
While you can use cash for transport fares, and get a new magnetic ticket each time you travel, it is better off getting a Suica/Pasmo/other IC and save cash for stores that don't take card.
At Haneda, if you head forward from the arrivals gate (after leaving the customs area), there are the train lines (Monorail and Keikyu) the right side and the bus ticket counters to the left. A number of the Keikyu line trains also run through the
Toei Asakusa Subway Line after stopping at Shinagawa.
There are a number of bus services from Haneda that go all over Tokyo, some direct to large hotels.
The hotel is very unlikely to let you in to a room before the published check in time, even if the room is available. The hotel will store your bags for you prior to check in, or if not, there are lockers at almost every train station. On busy days, such as weekends and public holidays when the station lockers are likely to fill quickly, the large major JR stations will also open manned baggage storage.