こんにちは Yinfon さん。
The only train that runs from Haneda Aiport into central Tokyo is the Tokyo Monorail. This runs through all the terminals of the airport (domestic and international) and some other stations before it arrives at Hamammatsucho. The only problem with this is I think the last monorail train departs just after midnight. So if you are arriving at the airport around 00:25 you will definitely not make it. (Plus you would still have to catch a cab from Hamamatsucho Station afterwards as other trains would have stopped by then.)
With buses, it looks like there is one Airport Limousine Bus which you can catch from the international terminal at 01:05am.
The last stop for this bus is the Asakusa View Hotel and arrives at 02:20am. This is the closest I can find to the Khaosan Tokyo Origami. (Probably a 10 min walk). You can see the distance if you can locate both hotels on a map.
From the looks of it, you need purchase the ticket from the 2F (closes at 01:00am, so giving you a minimum of 5 minutes to run from the counter to the bus stop).
Then catch the Airport Limousine Bus from bus stop 2 on the 1F.
The price for 1 adult is ¥1,860 (night bus price, and one way only)
If you are able to purchase the ticket and get to this bus stop before 1:00am you should be okay.
This is the website I looked on to find this information.
http://www.limousinebus.co.jp/en/platform_searches/index/4/244
If you want to search from the beginning to get to the link above go to:
http://www.limousinebus.co.jp/en/
> Then find the box that says 'Pick me up from the airport'
> Select From: Haneda Aiport
> Select To: Asakusa, Kinshichou, Toyocho
> Then finally select Asakusa View Hotel
Probably best to do a bit of further investigation as well, as I've never taken a bus from Haneda Airport to be familiar with the process. So I can't guarantee anything.
A cab from the airport will be pretty expensive especially when public transport is not operating. I would predict it to be approximately between ¥10,000 to ¥11,500 from the airport to Khaosan Tokyo Origami.
I would say hang around at the airport, but facilities pretty much close during the early hours of the morning as well. I think the shower facilities are open 24hrs and maybe the odd few convenience store.
Should you miss your bus, the next bus to Asakusa View Hotel is at 11:40am. Which by then you would be able to catch the monorail (starts running again at 5:17am), and catch trains/subways to Asakusa station.
I hope this help. Good luck too!