Aiiro (formerly known as Advocates) has long been a hub in Shinjuku-nichōme, and is a good place to get an evening started – thanks to the all-you-can-drink beer for ¥1000 happy-hour special from 6pm to 9pm daily. The bar itself is teeny-tiny; the action happens on the street corner outside, which swells to block-party proportions when the weather is nice.
Aiiro is welcoming to all and staff speak English. There's a rainbow torii (gate) out front; you can't miss it.