After you're seated, the bartender of staff will take your order and help you through the menu. It could be a few minutes if the bar is busy, but don't worry, they haven't forgotten or ignored you because you're a gaijin. You don't need to get up, wave, yell, hold some bills in your hand, or anything of the sort. Just wait! If there's clearly something amiss, then a simple sumimasen is the most appropriate way to get immediate attention from the staff.
Feel free to order from the menu if they have one and you're capable. But depending on what kind of experience you're after, this might not be the best way to order given that bar menus tend to appeal to as wide a variety of drinkers as possible. You came all the way to Tokyo, and you've made it this far. You want something special and unique, right?
Already have a drink in mind? Bartenders in Japan tend to have a massive quantity of recipes memorized, so they'll probably be able to immediately tell you if they are capable of making your drink of choice.
Don't know what to drink? Here's your chance to experience Japan's famed omotenashi culture if you haven't already. Have a think about what you like. Have a conversation with the bartender if possible. Did a bottle on the shelf catch your eye? Ask to see it. Ask if you can smell the aroma. You may even get a complimentary taste. What base do you like? Something with bubbles? What fresh fruits do they have tonight? Can the bartender offer a recommendation? Talk your way through what you like, what you don't like, and let that guide the bartender to come up with something for you.
Rather that paying per drink or opening a tab as you may do back home, in Japan, you'll just get a bill at the end.