This route is not a full destination ride as by itself it is less than 4 miles. You could hike it, but would probably just access the summit via the Appalachian Trail from 9d. This is an old carriage road, severely eroded, but a satisfying challenge on mtn bike.
Directions only are given for one way, but as it is an out and back, you will have no problem reversing the route to the start.
The easiest way to get to the start is via Metro North at the Manitou stop, which is only on limited trains on the Hudson Line. You must be in the last 2 cars of the train to exit at Manitou. You will ascend Manitou Station Rd, cross 9d (0.6 miles) where it changes its name to Manitou Rd. In o.2 miles it ends on South Mtn Pass Road. Take a right and ascend. In another 0.5 miles, or 1.3 miles from where you got off the train at Manitou Station (and a bit after the paving ends,) a poorly maintained turnoff will appear on the right. Opposite on the left side of South Mountain Pass Road you will see the markings of the Appalachian Trail as 2 white rectangles, one diagonally above the other indication a turn in the trail. Follow the turnoff to the right and cross around the gate, where the cue sheet begins.
The first section, to 0.6 miles, is a pretty near constant grade of 5% on a rocky, gravel road. After this, the route to the summit will undulate up on the ridge with some sections so challenging that all but the most talented rider will walk. Do not underestimate difficulty as a fall on this rocky surface surely will not end well.
You will not miss the turnoff to the viewpoint ending just beneath the summit overlooking Bear Mtn, the bridge, the Hudson and surrounding mountains. We commonly call the Anthony's Nose climb the ascent to the lookout on routes 6/202 south of the bridge, but the summit of this route actually takes you to the top of Anthony's Nose, the mountain, all 900' above sea level. The view is one of the finest in our region. The photo attachment juxtaposes a pen and ink drawing from an early 1930's edition of The New York Walk Book, that was done prior to the erection of the Bear Mtn Bridge, with a photo taken on a bitterly cold day in February 2016 [I don't think it went above 20 degrees F], obviously from the same perspective, both in winter. It goes to show that though the plots and actors vary, the scenery doesn't change very much.
If you have a realistic handle on your skills as a mountain biker, the amount you walk this route will vary up or down. Observing your ascent will give you important information you need to know about what to expect on the return.
You can also access this route from Peekskill via route 9 and South Mtn Pass Rd from the east, or Garrison/Cold Spring via 9d from the north. An option to fill the rest of your ride is to do Old Albany Post Road, an historic unpaved gem, taking it north, then down to Cold Spring via Indian Brook Road, also unpaved. To do this, go east on South Mtn Pass Rd to route 9, go north and take a right on Winston Lane, then a left on Sprout Brook Rd. The first left will be Old Albany Post Road. If mountain biking is the aim of your day, consider the Moneyhole Mtn Trail via the Moneyhole Mtn Access Trail off South Highland, and the optional spur to Chimney Hill, details in Garrison/Moneyhole Mtn/Sunken Mine/Stillwater/Cold Spring, in this same region's routes.