I'd say it is possible to do both in one day, but it would be a long day, you would be rushed, and you probably would not get to spend as much time as you want to on the Zugspitze. But you would get to check off two venues as having been there. I don't think the quality of you visit to the Zugspitze would be worth it.
I went up the Zugspitze once. I had lunch in Garmisch-Partenkirchen before taking the train and cable car to the summit. By the time I got there, the mountain was enveloped in clouds so that I couldn't see much.
The first train to Garmish-Partenkirchen leaves Munich Hbf at 6:32 and gets to GaPa Bhf at 7:54. That gives you 21 minutes to walk to the Zugspitzebahn Bhf, get your tickets, and leave on the 8:15 train, transferring to the cog rail train in Grainau at 8:30, and arriving at the top of the Zugspitze at 8:28. Or you can get off the cog train at Eibsee at 8:45 and catch the 9:00 cable car to the top at 9:10.
Tours of Neuschwanstein run until 18:00 (6 pm). You would need to pick up reserved ticket at the kiosk by 4:30 pm.
There are connections from Garmisch-Partenkirchen to Hohenschwangau leaving GaPa at
12:07,
13:00, and
14:04
getting to Hohenschwangau at
14:45 (train to Oberau, buses from there),
15:36 (bus with one 32 min transfer), and
16:18 (train to Reutte, 3 buses from there), respectively.
The first two options only give you about an hour at the top. You have to catch the 10:30 train, which gets to the GaPa station at 11:50, or take the 11:00 cable car which gets to Eibsee at 11:10, giving you 5 minutes to run and catch the same train at 11:15. If you can catch the 12:07 train to Oberau, you get to Hohenschwangau at 15:36. Or you can wait for the 13:00 bus (maybe have lunch while you wait) and get to Hohenschwangau at 15:36. That would be less hectic on both ends.
The last connection allows you an extra hour on the Zugspitze and gives you 12 minutes, which is enough, to walk up to the kiosk and pick up your reserved Neuschwanstein tickets. However, there are two 7 minute connections, to the bus in Reutte and a transfer in Vils.
There is the possibility of taking the 11:30 train from the top (or the 12:00 cable car) and making the 10 minute connection to the 13:00 bus to Hohenschwangau (15:36), but the Bahn does not freely offer that connection so they probably don't think you can make it. In that case you don't have to buy a ticket until you get on the bus.