The hotel bus is a good option only if you're staying at one of the hotels on the route (which tend to be the more upscale ones) or at a budget lodging within walking distance of one of the hotels.
Take the train, either the Keisei Skyliner if your hotel is in the northern part of the city or the Narita Express if it's in the southern part of the city. The Keisei Skyliner lets you off at Keisei Ueno, which is close to JR Ueno, which has good surface train and subway connections to the rest of the city. The Narita Express lets you off at Tokyo Station, which is also a major transportation hub.
You can easily find a "business hotel" for between $80- $100 per night, which is cheaper than you can stay safely in a lot of major cities. You'll get a small, plain but adequate room with a private bath and maybe even a breakfast buffet.
If that's too high for your budget and you're willing to risk having to sleep in a bunkbed in a dorm, consider a backpackers' hostel or the Hosteling International facility near Iidabashi Station.
But please, not a capsule hotel, which is likely to be full of drunks who missed the last train home.
If you don't already have a city guidebook, buy one. If you don't already have a map of the city, buy one.
Once you've reserved your hotel, find it on the map. Print out both the Japanese and English versions of your hotel's web page, the ones with the map (most hotels have maps on the page marked "Access") so that you can ask for directions if you get lost. (You will get lost. Japanese people get lost in Tokyo. But that's OK. Just ask where the nearest station "eki" is, and find your way from there.)
For "old Tokyo," visit the Yanaka neighborhood, the Edo Tokyo Museum, and Sengakuji, which is dedicated to the 47 Ronin. The Shitamachi Museum near Ueno Park shows life in that part of Tokyo before World War II's fire bombs leveled it. The National Museum in Ueno Park is kind of dingy, but it gives you a good survey of pre-modern Japanese art beginning with the earliest archeological findings.
I recommend the view from the 45th floor of the Metropolitan Government Building (subway stop: Tocho-mae on the Oedo Line), and not only because it's free.
Cheap eats are to be found in the little mom-and-pop restaurants and on the restaurant floors of department stores.