Which way is the best to get there is highly depending on where you are coming from. Since Tsukiji is very close to Shimbashi, you can use the Yamanote Line to get there and then walk. The first train arrives there around 4:30. From Ueno for example, at 4:45. Hyperdia offers a search engine in English on this.
If you want to stay in an internet Cafe, there is a map of all MangaKissa shops around Tsukiji. Finding the right spot is tricky. Not all places offer flat beds, and some are open only to members. Also, without speaking Japanese, you will have issues using these places. There might be also registration fees.
This one here seems to be open to non-members (but members can get some discounts), have a flat seat and is quite close to Tsukiji, again at the Shimbashi train station. Google also translates this page.
It will be hard to find anything closer since the Tsukii area is not considered a major train station and therefore not the best area to build these shops. However, if you are willing to walk 30-45 minutes, you can also look around Ginza and Shidome.