Izakaya are fantastic. There are also quite a few “one coin bars” around the city. There are a few in major areas like Shibuya and Shinjuku or even Ebisu. They will literally have a sign that says “one coin bar” and it’s easy to ask anyone because they all understand this term. These places generally have drinks at ¥500 –which is quite cheap in Tokyo, trust me.
Alcohol is readily available in many vending machines (though you might need to swipe id at some) and all combini will have beer if not more (many places will sell wine, chuhai, sake and more).
Some restaurants have a set deal if you have 4+ people eating where you usually pay 3000 Yen (and up) per person and get a set course and all you can drink for 90 minutes. Also if you hit the clubs where foreigners hang out, many have an “all you can drink all night” special for one fee. Keep this in mind when you’re enjoying the night life!