Description: Here it is, the proper cocktail bar. This is what you came to this site for, right?
System: The Tokyo Bar FAQ should answer any questions you have about the typical setup for a cocktail bar in Japan. But to summarize: you sit down, order whatever, and pay at the end. Simple!
Budget: It's difficult to generalize here. You can pay as little as 100 yen per drink at some places, or 10000 yen for 30ml of cognac at others. Most typical bars tend to the 1000-1500 yen per drink range. You'll almost always get charged otōshi or a seating charge of some sort.
Making friends: Again it's difficult to make a rule of thumb for this aspect when it comes to bars. Some clearly encourage customers to fraternize among themselves to meet new people. Even so, bartenders generally keep a watchful eye to make sure female customers are comfortable. They will ask male customers to dial it down if they feel they're coming on too strong. With the notable exception of hotel bars, the higher in grade you go, the more unlikely it is to be okay to chat up strangers.