I've been to every major sumo tournament at least once (most more than once), and sat in both boxes and seats in Nagoya (as well as other tournaments). While the answers here are good, there are some quirks that more casual fans may not know or appreciate.
Matches start early in the morning (8:30am) from the lowest ranked wrestlers. The main broadcast is from 4pm-6pm, and that is the main event with the highest ranked wrestlers. After the final bout there is a lower-ranked wrestler who spins a bow around, and a Japanese drum plays everyone out of the venue. Most people who watch live in the good seats arrive around 2-3pm to catch the two highest divisions. Hardcore fans will show up in the AM. Since you can eat in the venue (you can bring food, and they sell it there) I usually show up with a lunch around noon.
It is perfectly okay to sit in the best seats open until around 2-3pm as long as nobody is in them. Even if you get the cheapest ticket available, you can watch several hours of sumo from the best seats in the house for absolutely no additional charge. If you don't care about seeing the top ranked guys up close, then buying the cheapest seats will still get you to see a lot of sumo.
Additionally, the earlier matches are much quicker. There is a match every 3 minutes or so for the bottom 4 divisions, which becomes about 4 minutes for the second ranked division, and 6 minutes each for the main event. The lower ranked matches also tend to have a lot more crazy results as the people are younger, less practiced, and far more athletic (you may see wrestling throws, flips, and other crazy stuff that doesn't work in the top division). Depending on what part of sumo interests you, you can decide which part of the day is best to see, and base your ticket choice on that.
The prices of seats are very different depending on where you sit. If you look at the official site for a tournament the prices don't show everything. For instance, the best seats are shown as 桝A席 (Masu A Seki) for boxes, but what they don't show is that the boxes from Row 7 and up are the only ones available through the site. The better boxes are bought by companies which sell tickets including souvenirs for a huge markup that can be 1.5 to 2 times the price listed on the site.
If you want the best seats in the house, expect to pay 20,000-30,000/person. For normally available boxes through the net (the link above) expect to pay 12,000 JPY/person for a box seat. If you want the cheapest reserved seats, expect to pay 3,000 JPY/person for a back row seat.
The Nagoya tournament this year will be from July 13th (Sun) to July 27th (Sun). The 3rd Monday (July 21st) is a national holiday, so the middle weekend is a 3-day weekend. The highest ranked wrestlers in the highest division face off against each other the last days of the tournament, so the last weekend (especially the last day) is most popular. In Nagoya, on the last day, the fans descend on the ring in the center and disassemble it (they bring pickaxes and the like, and really take it apart to bring home souvenirs). This is purely a Nagoya thing.
Here is the popularity by day:
- Top: 15 (Sun), 14 (Sat)
- High: 1 (Sun), 8 (Sun), 7 (Sat), 9 (Mon)
- Low: Weekdays (2, 3, 4, 5, 6, 10, 11, 12, 13)
There are 4 sides to the arena, 正 (Front), 裏 (Back), 西 (West), and 東 (East). The best seats are 正 (Front) because they mirror the angle on TV and you get the clearest view of the action. If you get a seat in 裏 (Back) then the judge who stands between the two wrestlers will block your view of the action at the very start. If you pick 西 or 東 (East or West) you are looking at the back of one of the wrestlers who is even bigger than the judge. Try to get a seat in the front section if you can. The same-day tickets are 自由席 (no assigned seating), so this only applies if you are reserving seats ahead of time.
If you want to reserve a good seat on a top/high day, you need to call the instant tickets are available (they sell out within about 10 minutes -- to get them you have to use the net and call on at least two phones and pray you get through). Tickets usually go on sale on a Sunday from 9am (JST) about 6 weeks before the tournament starts (note: the reservations will all be done in Japanese). The day they go on sale is announced on the website. If you just want cheap seats on a weekday, you can probably show up the day of and purchase them at the door (2,000 JPY/person).
Going Alone/With Others
If you are going there alone, boxes are awful. They are cramped and you get stuck with people you don't know. I highly recommend a seat, or, for the daring, trying to reserve a 溜席 (Tamari Seki) which are the cushions on the floor surrounding the ring. You aren't allowed to eat or drink in those seats, but you get a great view, and they cost similarly to the boxes (depending on the row, you may be asked to wear something over your clothes so as to not distract the wrestlers -- the brown thing in this photo).
If you are good with people and generally friendly, then you may have luck getting a seat in the back, sitting up front early, and being invited to sit with folks in boxes who have a spare seat by merit of being foreign. This is not a high-probability event in Nagoya compared to Osaka, Fukuoka, or Tokyo, but you can give it a shot.
If you go with someone else, there are pair seats available at some tournaments, but are likely very popular and will sell out quickly (they tend to be very few). Some tournaments (Osaka comes to mind, Fukuoka too I believe) have a few single-person box seats but those are super popular and usually sell out incredibly quickly as well.