Soak
Before attempting any other type of cleaning method, soak your antique glass bottle in a solution of warm water with a few drops of dish soap. Spread a dish towel on the bottom of your sink before preparing the water. As you fill the sink, the towel will want to rise due to air bubbles forming underneath it. Pat the towel with your hand to remove the air bubbles, and once they are all out, the towel will sit flat on the bottom of your sink. This will cushion your antique bottle from the sink's surface.
Be mindful of the water temperature before you put your antique bottle in the sink to soak. You don't want water that is significantly warmer or colder than the bottle itself, as this could cause the bottle to fracture.
Allow the bottle to soak in the solution overnight. Rinse thoroughly with warm water and allow it to air dry.
Inside
If the inside of your antique glass bottle didn't come clean via the soaking in the sink method, there is another step you can take. Purchase some denture cleaning tablets at your local pharmacy or grocery store. Fill your bottle with water that is at room temperature. Break the denture cleaning tablet so the pieces will fit inside the bottle neck. Insert them into the bottle, and allow it to soak overnight. Empty the bottle the next day, and rinse with warm water.
Last-Ditch Efforts
If the previous steps haven't left your antique glass bottle sparkling clean, there are two more steps you can take to remove stubborn stains.
For rust or other hard-to-remove stains on the outside of your bottle, purchase some muriatic acid at your local hardware store. Wear rubber gloves when handling it. Dilute the muriatic acid using one part acid and four parts warm water. Dip a nonabrasive scrubbing pad or sponge into the solution, and gently rub the outside of the bottle to remove the stains.
For stains inside your antique glass bottle that haven't yet lifted, try this solution. Purchase some liquid paint stripper at your local home and garden supply store or hardware store. Fill the bottle with the liquid, and seal the bottle with a piece of plastic wrap secured with a rubber band. If the bottle has a cork or cap, secure this instead. Leave the paint stripper inside the bottle for up to three days. Remove and rinse thoroughly with warm water.