The cheapest way to ship a poster is to keep your package within the max weight and dimensions of USPS First Class package.
USPS First Class package
Max weight: 15.99oz
Max dimensions: 22x18x15"
For tubes, the height and width will be the same and the rate is determined by the weight of the package.
For USPS Priority Mail
Max weight: 70lbs
Max dimensions: Length + Girth < 108"
Please note- you cannot ship tubes via Priority Mail Cubic!
Priority Mail is zone-based pricing. That means for certain zip codes you will have to enter the dimensions of your tube in order to obtain the correct rate. Entering dimensions is always recommended to ensure your packages receive the correct rate from the postage provider. Incorrect dimensions can result in packages being returned, or rate adjustments applied to your postage provider account after shipping.
Check out USPS's Dual-Use Tubes
These are free 'tubes' that can be used for Priority Mail or Priority Mail Express: https://store.usps.com/store/product/shipping-supplies/dual-use-priority-mailpriority-mail-express-small-tube---1098s-P_O_1098S