A tour of both castles is called a King's Ticket and can be obtained online on the Hohenschwangau website.
You must pick up your reserved tickets at the Hohenschwangau kiosk one hour before the start of your first tour. I've never done both in the same day, but I understand your first castle tour is of Hohenschwangau (~40 min), which is the closest to the kiosk, then you have about 45 min to get up to Neuschwanstein for that tour (also ~40 min).
Since you are doing this on a weekend, you can use a Bayern-Ticket for the train and bus and leave before 9 AM. It you catch the 7:53 train from the Munich Hbf, you get to Füssen Bhf at 9:54 and to the Hohenschwangau bus stop at 10:13. It's a short walk up to the kiosk; you should be there by 10:30. The line to pick up reserved tickets shouldn't be that long; you should have them by 11, so reservations starting at 12, noon, would work. But just to make sure, make them for 1 PM (1300). You can use the extra time to have lunch in the village and to visit some of the shops before walking to Hohenschwangau castle.
Both tours and walking to Neuschwanstein should take around 2 hours, so you'd still have time to take the bus (also covered by the Bayern-Ticket) into Füssen and see that attractive town before heading back to Munich.