You can pick up free wifi codes for travelers at one of the NRT terminals and some other info desks. All the JR and subway stations in Tokyo now have free wifi, and some (not all) cafes and other places do as well. A lot of "free" wifi requires email signups (targeted at a local audience since local phones come w/ email). Also, check this app out, this should get you access throughout most of Tokyo: http://wi2.co.jp/tjw/english.html

IMO, it's worth the cost to either rent a pocket-wifi or to buy a prepaid data SIM: https://randomfoo.hackpad.com/Japan-Notes-ecTyWVuzaQD#:h=Mobile/Internet

(I always pick up local SIMs but I'm also now carrying around a US T-Mobile post-paid data SIM which has unlimited 2G/3G roaming in 120 countries. It's pokey, but good in a pinch for maps/comm/translation)

There are some incredibly laid back cafes w/ power/wifi, which is probably a better option than co-working spaces. This might help: http://tokyocheapo.com/business/internet/free-wi-fi-cafes-in-tokyo/

Right now pricing for lodgings is through the roof because of sakura, and then golden week later this month. You can do a capsule hotel, especially if you're going to be wandering around during the day (like someone else mentioned, you usually get kicked out for a bit during the day). There's actually a fair % of people that live in capsule hotels permanently (also capsule hotels aren't the cheapest options anymore either, you can overnight in some internet cafes as well). There's a lot of info online so you can do your own research. Depends how cheap and choosey you are, I guess. There are also better/worse capsule hotels. The one I stayed at in Nagasaki was a lot more pleasant than the one I stayed at in Tokyo.

If you're really budget constrained, you may look for some places out of town (like west towards Mitaka, east in Chiba, or south in Yokohama) as places that will let you easily visit Tokyo (30-45m on the train) in the day, but that should be significantly cheaper to stay at.

If you've wanted to visit Japan, I highly recommend it. It's a very neat/unique place (even as someone that has spent a lot of time in other parts of East Asia), and while it's a lot more expensive than SE Asia, both the difficulty and cost are usually way overestimated (for example, although housing was more expensive, due to the huge amount of cheap (and delicious) food options, my avg cost of living ended up about the same (slightly lower even) in Tokyo vs Berlin last year).