Even if you can't find the cache, the view is amazing. This path is lit at night and worth going for the view, but you'll probably need daylight to find the cache.
The nearest MRT stop is Taipei City Hall. The path entrance is about 8 blocks away. You can walk from the MRT to the 7-11 at N25.01.610 E121 34.111
The start of the path is at N25 01.650 E121 34.248.
Climb up to the top of Elephant mountain. Enjoy the view for a bit and then follow the trail back towards Muzhi Mountain. The cache is down the stairs, and before you start heading upwards again. The cache is stuffed into a opening on the side of the cement path and covered with some rocks.
(By the way if you keep following the path where this cache is located, you can walk for another ~2 hours to the top of Muzhi Mountain. It's really a great hike, with a great view. Make sure you bring plenty of water and daylight if you're going to do the hike all the way to the top of Muzhi Mountain.)
Bug spray may be a good idea in the summer months. There can be some mosquitoes.
Small note on cache maintenance: My geocacher friend "kctwu" in Taipei will watch the Elephant Mountain Geocache that I placed. He lives in Taipei Taiwan and knows the cache location.