Overview:
The Hohe Geige (3395m) is the highest peak in the Geigenkamm between Pitztal and Ötzal. It features a nice climb and a very good view.
You can do it at one day, but i suggest staying a night at the Rüsselsheimer Hütte.
The way up to the refuge is approx 1-2h. From the Rüsselsheimer Hütte to the summit using the west ridge 3.5 to 4 hours and the normal way 3 to 3.5h. The west ridge is by far the more beautiful way.
Down to the Rüsselsheimer Hütte in approx 2.5h, back down to the parking place in 1h.
Tips:
You have a good chance to spot ibex on your way up in the morning after leaving the Rüsselsheimer Hütte.
The Rüsselsheimer Hütte is the perfect refuge for this trip. It offers space for about 30 people and is a very charming refuge. It's lying at 2323m, it takes something between 1h and 2h to get up to it from the parking spot.
Here the way branches. Follow the north way to the western ridge, the marker shows "Hohe Geige - Westgrat". The west ridge is the more beautiful path. It difficulties are mostly UIAA I, a few spots II. To the east you can follow the normal path, but better descent using it.
Enjoy the wonderful view. Ortler, Bernina, Weildspitze, Weißkugel, Hintere Schwärze, Similaun, just to name a view summits you can see. There is nothing blocking your view here, if you have luck with the weather, it's just amazing.