Tokyo metro subway is not under operation around 3-4 a.m. You need to catch a taxi to the market.
You can catch a taxi on the street or at taxi stand. Just hands up to catch a taxi. The taxi doesn't stop while the other passenger is on. You can just stand beside the taxi at taxi stand. All taxi in JP automatically open the rear door, so be careful not to be stand closer. You need to sit on the rear seat. If you are a group of 4, one passenger needs to sit on the passenger seat in front.
It will cost you 2,500-3,000 YEN between Asakusa and Tsukiji market. All taxi will charge an additional 20% night fare between 22:00 - 5:00. It takes about 20 mins to Tsukiji.
All taxis in JP have a charged meter that you can read. No advanced negotiation before on board.
I strongly recommend you to ready some 1,000 notes in advance if you use a taxi. You can use 10,000 notes but the most drivers don't like to accept a large amount note because of security reasons. No tip is necessary but you can leave a tip if you like.
You can order a taxi in advance. Ask at the accommodation desk that you need a taxi early in the morning. The service charges you 400 YEN for a taxi order and another 400 YEN for picking up at the time you have ordered, that cost you additional 800 YEN.
Those taxi drivers in Asakusa area know that foreign tourists visit Tsukiji early in the morning so it should work well, just say "Tsukiji-shijo", or write or print "築地市場" and show it to the driver. If you order a taxi in advance, you need to tell the operator where you want to go in advance.
I think you should be there by 3:30 a.m latest(2:30-3 a.m preferred) if you want to see the auction. There are many tourists want to visit the auction. The attendance is competitive throughout a year.
You can bike to Tsukiji but I don't recommend unless you are a frequent Tsukiji commuter. It's still freezing at night in Feb and Mar.
There are many police officers stand and ask you to stop and check your bicycle number especially those bicycle riders at night. It will be nasty experience for you if you caught by police.
There are a large bicycle stand under Tsukiji-shijo station. You are not allowed to park your bicycle at prohibit area(the most area in Tsukiji market is prohibited).
Read these carefully and print if you need.
Tsukiji and the other markets are not tourists spot, it's trading market where a large number of traders work. You need to follow the regulation.
http://www.shijou.metro.tokyo.jp/english/market/tsukiji.html http://www.shijou.metro.tokyo.jp/pdf/english/tsukijiguide.pdfTsukiji market calender is available here.
http://www.shijou.metro.tokyo.jp/calendar/Tsukiji closes all Sundays, the most of Wednesdays, national holidays, and special days set by the market.