Advertise in local or national newspapers. Depending on the scope of your production, that newspaper might be the New York Times or the local Pennysaver. But you'll be able to reach a vast potential audience this way.
Build a website. No matter how limited your budget, pay for a website, and include still pictures, bios and a synopsis of your musical. Include a short video or audio teaser if you have one and a link to purchase tickets in advance.
Design an attractive flyer or postcard announcing the musical, complete with performance dates and information on how to buy tickets. Include a link to the musical's website. Hang the flyer on supermarket bulletin boards and other places where people congregate, or have college students or cast members pass them out on busy street corners.
Get your stars on radio or TV shows. No matter how small your immediate market, there's always a radio or TV show or podcast that will be glad to feature your show's stars or producer talking about the upcoming performances.
Tell everyone you know about your musical. You never know who they might know and it could turn into extra publicity for your production.