The best way to approach the summit block is probably via the southwest ridge and the saddle to the southwest of City Mountain. We approached this ridge from the Nipple Peak Trail, but you could also access routes to the ridge and saddle by using FR 488 and FR 489. The "standard" route up from the Nipple Peak trailhead requires about 1500 feet elevation gain. The FR 488 and 489 route has a bit less, but more ups and downs and would be a long drive on rough roads (if they are even open).
The south face makes a pretty good descend route, but going up hill, the routefinding might be easier using the southwest ridge.
The summit block and ridge of City Mountain is technical. There are a couple of 4th class/low 5th class routes up to the summit. One is on the east side of the summit block. The other reaches the knife edge just west of the true summit and from the north side.
The summit of City Mountain.
From Steamboat Springs, drive to the north end of town and to where County Road 129 leaves Highway 40. A sign marks "Clark" and "Hahns Peak". This is the same road heading to the airport. Follow County Road 129 north for about 32.2 miles, driving past Clark, Hahns Peak village, and Clark. 0.2 miles before the trailhead you will pass the Summit Creek Forest Service Station, but it doesn't seem to be staffed often. 0.2 miles past the ranger station, you will come to a sign for FR 47 and the Nipple Peak Trail. If you have high clearance, you can drive west a short distance to a fence. I would not recommend driving beyond the gate as the track is usually muddy and in very poor condition.
Route and Trailhead Map.
From the trailhead, follow the old Jeep Road west. There are some big mud holes and I wouldn't recommend driving the track. The track is closed in summer anyway.
The trail isn't heavily used, but it is easy to find and climbs gradually through an alternating pine and aspen forest.
Along the Nipple Peak Trail.
After just over a mile, the trail trail reaches a little creek and follows it for a while. There is a signed junction here. The main trail continues straight and is singed for Nipple Peak and Slater Park. To the right is an old mining track.
To climb City Mountain, follow the old mining track to its end and then climb the slopes on the left. You want to contour over to the saddle southwest of City Mountain. There is quite a bit of deadfall and you want to stay above one landslide on the traverse.
City Mountain as seen from the approach.
If you ascend a bit higher than the landslide, you might find a game trail that makes things easier. Continue to contour over to the saddle where you will find a few game trails.
You want to climb directly up towards City Mountain by following the indistinct Southwest Ridge. The biggest obstacles are steep slopes and deadfall. Continue up the slopes until the forest disappears in a boulderfield.
While avoiding the loosest of the scree, climb through the boulderfield on the right side and up to the summit block, which is visible once you are above the trees. You wan to either climb up to the left side of the summit block or ascend over the summit block not far right of its west terminus via a class 3-4 scramble.
Class 3-4 scramble.
Follow the north side of the summit block/ridge all the way to the east side. From the east side of the summit block/dike, climb up the semi-technical knife edge ridge until you can touch the true summit. The climb isn't that difficult, but there is some exposure. Although the rock is somewhat solid, some of the blocks on the ridge are severely overhung, so use caution!
The summit ridge of City Mountain.
Once you are on the summit, you can return the same way, or descend the South Face from near the west terminus of the summit knife edge.
Because the going is rough throughout much of the climb, the climb should take half a day and has about 1500 feet of elevation gain.
Map, compass or GPS, and a good paid of boots or shoes. Sticky shoes can be very useful. If you are afraid of heights, a rope and a small rock rack can be useful. A helmet is recommended as well.