Once you've had your fill and it's time to head out, you'll need to settle up the bill. Once again a sumimasen will work fine here to get the staff's attention. You can ask for the bill by simply saying okaikei. If you can't get that out for some reason, just make a small X by crossing your index fingers: this is Japan's equivalent of the "writing" gesture used to get the check in other countries.
Cocktail bars in Japan rarely provide itemized checks, nor can they split across multiple credit cards. Generally you'll receive a small piece of paper with just a single number written on it. This is your bill and is inclusive of all required charges and taxes. It may be higher than expected -- see "What is otōshi?" for more details.
Tipping is neither required nor expected. Just pay the amount on your bill.
In Tokyo, you should be able to use credit cards to pay at the vast majority of bars. Even so, it's always safer to have cash on hand just in case. When in doubt, ask if it's possible to use credit cards before ordering your first drink.
Whiskey Richard is the founder of nomunication.jp, and has been saying he has lived in Japan for ten years for the past four years. He enjoys drinking basically anything except straight-up Campari.