If a restaurant typically makes someone start as a server before bartending, should I put that on my job app.?

You should express that you want to move up. Many companies love to get people who want to go somewhere. Let anybody and everybody who would have an opinion know that you want to move up and show them you are ready for it.

Go ahead and fill your app, selling your qualifications as a bartender. Personally hand your package to the hiring manager. Make a strong case for the bartending position, as if you never heard the rumor that the bartenders come up from the servers. He may take you on for the post you applied for. But before you hand the application in, think seriously about whether you're willing to start somewhere off target, then rise up to your goals. Know where you stand, and if he insists on the rumored policy, you can be ready to switch gears and sell yourself as a server. He'll probably appreciate your enthusiasm and mark it as a plus towards bringing you on board! Supposed you're hired, but not behind the bar. That's okay! Even if you're a dishwasher, you're in a much stronger position on the inside than from the outside. Excel at your job, network, and look for the next chance to get where you want to be.

Absolutely. Tell them where you would like to go even if it's not possible to start there in the beginning.

Nah, bar tending is all you need to put on it... Though, I absolutely agree with the top person... : )