At http://ch.nicovideo.jp/shogi you can find a schedule of upcoming live games on the NicoNico channel. They're all in Japanese. Also, unless you're a premium member (costs 480 Yen/month) NicoNico will probably kick you off any popular game. You can get a taste for it without being a premium member, but it's too annoying if you're not one to really use. You can always watch a game up to one week after it's complete, sometimes longer. They have the title games from all the big tournaments except Oshou, a variety of other significant games (e.g. any game where 14-year-old Fujii is playing), and all the games from the entertaining Eiousen tournament (1 hour time controls; the newly 8th major tournament), for which I think they are a major sponsor. Games usually start at 9 or 10am Japanese time. If you're watching some completed game that took 12 hours or something like that, you can easily scroll forwards and backwards to get through the game more quickly, and still see the key moments.
At igoshogi.net Premium, you can also watch live, or recently played games, again with Japanese-only commentary, but again it's a paying service. Silver membership costs 1000 yen/month or 10000 yen/year and gold membership costs twice that. Gold members can watch live games, silver members watch archived games (they show up a couple weeks after the actual play, and are archived for 3 months). There are also instructional courses, again all in Japanese, broken up into around 30 videos of 10 minutes each. The main tournaments covered in igoshogi.net are the Ryou, Oushou, and GinGaSen tournaments.
The scrolling feature on igoshogi.net doesn't work as well as NicoNico, as the video has to have streamed up to the point you want to scroll to, so I tend to watch longer games on NicoNico rather than igoshogi.net.
The GinGaSen games on igoshogi.net are particularly nice because that tournament has a time control meant for TV (similar to the NHK cup), and the games are lightly edited to make sure they last 90 minutes. There are two GinGaSen games per week; unfortunately, the GinGaSen games available lag behind the real tournament by a few months, but at least they're archived for several months, and they also show plenty of good vintage GinGaSen games from previous years.
By the way, you should be careful to sign up for igoshogi at the beginning of a month. You'll pay for the entire month regardless of which date you sign up on. Also, although their sign-up is hostile to non-Japanese speakers, you can do it; just enter Tokyo as your prefecture and put in some Hiragana for your name using a Japanese keyboard when they ask for that. So long as you have a valid credit card, you can sign up. In general, read their FAQ carefully to see whether you want to sign up.
I have a NicoNico premium membership and and IgoShogi Silver membership. That way I can certainly watch plenty of exciting Shogi. I'm only a Japanese language beginner so I don't understand much of the commentary, but for me, it's still fascinating to watch, as I can understand the positions, and when a variation is given by the commentators on their commentary boards, I can follow that.
I also like buying Shogi Sekai magazine 将棋世界 magazine on amazon.co.jp. They ship abroad. Just do a search by copying and pasting the Kanji I wrote. Again, everything's in Japanese, but lots of pictures, cross-tables, puzzles that are easy to follow for non-Japanese, and you certainly get the latest news, as this is the official magazine of the Japan Shogi Association. It's 800 yen per issue, but then shipping abroad adds a fair amount, so I try to make multiple purchases at one time to keep the shipping costs reasonable.
If you want to see what one of these magazines is like, you can get the Shogi Sekai app for iOS, which gives you one free old issue from 2010. They originally intended to sell the magazine by this app, but discontinued that in 2014, and you can't buy the back-issues that way either. I much prefer the paper version anyways, but the app still gives you one sample issue to look at.
Finally, the Shogi Live app for tablets and smart phones gives you many recent games (from the last 6 months) that you can play over, with Japanese commentary. No video, just an app that gives you the moves as they happen, and then stores the games for about 6 months. Probably about 1/2 of the pro games show up, including all the most important ones. The service costs $4/month.
I think there is an app Shogi Live. You can also become part of World Shogi News group in Facebook. They share games sometimes and you can ask questions.