This is a great question; I'm travelling to Japan next month and I'm tempted to add this to our itinerary. Codinghands did a great job of covering the basics. I read Japanese, so I can clarify some details.
There is an English-language overview of the Metropolitan Area Outer Underground Discharge Channel tour on the official website. It has links to sub-pages which give a lot of other information.
The tour starts from a building called the Ryukyukan (龍Q館), Showa drainage pump station. The address is 720 Kamikanasaki, Kasukabe city, Saitama 344-0111.
Getting there by train: from central Tokyo, take JR to Ōmiya, then change to the Tobu Noda line to Minami-Sakurai Station (南桜井駅). This should take just over an hour. From there it's 3km to the Ryukyukan, a 7-minute 1000-yen taxi ride or a 40-minute walk. Getting there by car: from the Tohoku Expressway's Iwatsuki Interchange, take Highway 16 towards Noda for about 17 km (30 minutes). From the Jouban Expressway's Kashiwa Interchange, take Highway 16 towards Noda for about 20 km (40 minutes). See the official Access Map for details.
The tours are 90 minutes long. About 10 minutes of that is in the "Underground temple", apparently. There are about 100 steps to climb. It is offered to ad hoc groups of 1-25 persons on Tues-Friday, at 10:00, 13:00, and 15:00, and to groups of 26-50 people on Mondays three times a day. The tour is in Japanese only, so each group with non-Japanese speakers must have at least one person who can translate safety instructions. If you come 30 minutes early, there is an 8 minute video in English which they can show.
Note that parts of the tour may be cancelled if, say, the tanks are full of floodwater. Children under the age of 5 are not allowed. Children from 5-19 years are allowed if accompanied by at least 1 adult per 5 children.
The tours are free, but prior reservations are required. The tour slots do fill up, so booking a few days in advance is helpful. You can book on the internet (Japanese language pages) starting 28 days before your desired tour time, ending 7 days before; or by phone on 048-746-0748 (in Japanese) from 28 days before to the day before. They want to you to arrive at the tour reception 30 minutes before tour time; latecomers will miss out.
For questions and to make reservations, call 048-746-0748 (in Japanese).
Other sources of information:
P.S. "首都圏外郭放水路 見学会受付" means "reception (desk) for the tour of the Metropolitan Area Outer Underground Discharge Channel". The term 見学会, which I translate as "tour", is ken gak-kai, or "see-learn-meet", which I think is a marvellously evocative term.