From CR344, drop south to an area that offers some camping. The road makes something of a circle here. In 2005, we had to drive through a long, muddy beaver pond before doing a low water ford of the main stream. After crossing the stream, the road narrows quickly and enters aspen forest. This road is identified as FR348. It was at the 5th switchback that we could not get our pickup around and at one point, found ourselves rocking back and forth on two tires because of the uneven character of the road. As mentioned before, if you don't have a short bed, high clearance 4WD, it may be best to begin hiking after crossing the creek, if you can find a place to park out of the way.
Hike or drive FR348 all the way to a pass/saddle at 12,600 feet where a cabin/structure is located now. (Perhaps the road has been improved in order to get materials & supplies to this cabin?) It's 5.25 miles to this point. The saddle is above treeline. You can count the switchbacks on the way up to amuse yourself. From the pass, proceed east along the ridge and hike over or skirt around one point of 12,980 ft., drop to a saddle, then go over Pt. 12,842. From that point, follow the ridge as it turns to the NNE and walk as far as marker 12,660. To this point, the majority of the hiking has been on tundra with short stretches of rubble. At 12,660, make a decision. Either continue on the ridge NNE to intersect the west ridge of UN 13,626 or, cut east across the head of the basin for Grouse Canyon, losing 200 - 300 feet in elevation, then hike steeply up ENE to the summit of UN 13,626. If you continue NNE to intersect the west ridge, expect more rubble, plus a ridge traverse with lots of little ups and downs, but never exceeding Class 2 work.
If you chose to drop eastward into the Grouse Canyon bowl, you can avoid most of the talus by losing about 250 feet in elevation. This will keep the route on tundra and tundra may be followed almost all the way back up to the summit of UN 13,626. Neither route will make much difference mileage wise. From the summit, enjoy a different perspective of Mt. Princeton, one which most will never enjoy. For the return, go back as you came.
If you have 4WD that can make it to the pass, then the hiking mileage one-way to UN 13,626 becomes 2.35 miles with 1,670 feet of elevation gain, plus perhaps 600 more on the return.
Links to other information, routes & trip reports for this peak that may be helpful.