Visiting Neuschwanstein and Zugspitze from Garmisch

That doesn't sound very realistic to me.

Here's a map of the area's railways. Füssen "West" station is at the tail end of a trunk line. But you can't get there by train alone very directly from Garmisch. And once there, you must take a shuttle bus to the N'stein stop.

It's perhaps 2.5 hours by train + bus + bus from Garmisch to N'stein via Reutte and Füssen. You may be able to find some bus-only journeys instead... You can look at the DB website yourself for these options and for the exact timing at the hour of day you think you might be there. Better check to see whether you can get back to Garmisch as well sometime later that day if you do this.

The Zugspitze overlooks Garmisch and you can take the cable car or cog train.
You will need transport to Neuschwanstein, either take a tour or look for transport.

A taxi for the Zugspitze would be easy.

I think either the Zugspitze or Neuschwanstein are doable in a day, but doing both would be a rush. If you spend enough time on the Zugspitze to justify the cost of getting up there, it would be hard to come down and get to Füssen in time to see the castle.

BTW, the fastest way to the castles is probably to take a bus (RVO 9606, about 2 hours) from in front of the Garmisch-Partenkirchen Bahhnof to Hohenschwangau via Oberammergau. The last bus back to Garmisch via Oberammergau leaves Hohenschwangau at 5:18 pm. After that, the only way back is a bus at 6:50 pm to Reutte where you can catch a train to Garmisch-P.