This is one of my favorite hikes. So check it out if you are in Appenzell.
How to get there: Take the cable car from Frümsen up to Staubern. From there You have a nice hike of around 1.5 hours until you reach the actual Saxer Lücke. Parts are a ridge walk, and you have amazing views along the way. If you are low on time, you can easily walk back from there and go back down from Staubern. If you have more time and want to explore more of the area, keep going down to the Fälen Lake and Sämtiser Lake. Your final destination will be then Brülisau from where you have to take a bus back to your car.
Price: a return ticket is 36chf, single ride is 18chf. NO REDUCED FARES.
Tip: This is one of the very view cable cars which is running until late. So you can even watch sunset from here.