There are numerous walking and biking routes in Shivapuri National Park, and Nagi Gompa makes a good place to start. The rough road contours from the nunnery all the way along the southern side of the park to its easternmost point, offering various ways to drop down to the valley. One good, bikeable route descends from a fork some 500m east of the nunnery, following a lovely ridge line south to Gokarna or Kopan; this is also an attractive (if long) route up to Shivapuri peak.
From the Tamang village of Mulkharka, 10km east of Nagi Gompa, along the road, walkers can descend to the small town of Sundarijal in less than half an hour; from here plentiful buses run back to Boudha and Kathmandu. Alternatively, you can trek northwards and upwards through the middle of the park from Mulkharka (where there’s basic accommodation) to the pass at Borlang Bhangjang (roughly 3hr), from where you descend to Chisapani, at the park’s northeastern side (1hr); this is traditionally the first day of the Helambu Circuit.
From Mulkharka, you can also take a classic mountain-bike route, heading east on the rough road all the way to the park gate at Jhule, and continuing south from there to Nagarkot or Sankhu; you could even turn north and carry on all the way round the fringe of the park to Chisapani. Budhanilkantha to Chisapani would be an all-day ride of 30km (count on 8hr). Between Mulkharka and Jhule, walkers can descend directly south to Sankhu, via the Bajra Yogini temple.