Technically, Arima Onsen ''is'' a part of Kobe, but it's on the other side of Mount Rokko and thus a separate city for all practical purposes.
The '''practical way''' is to take the Hanshin, Hankyu or Sanyo Line to Shinkaichi (_V_J_n) just west of Kobe, then transfer to the Kobe Railway Arima Line to terminus Arima Onsen. (Note that express services terminate at Arimaguchi, so you'll need to switch to a local train.) Alternatively, if you're coming in on JR, get off at Kobe's Sannomiya station, take the Kobe Subway to Tanigami (_J__), and connect there to the Kobe Arima Line. The whole trip will cost around ¥900-1000 and take about half an hour.
The '''fun way''' is to take the Hankyu line to Rokko station and continue up Mt. Rokko on the cablecar (10 min, ¥570/1000 one-way/return). Connect with the bus loop line (10 min, ¥250) to the Rokko Arima Ropeway, then take the ropeway down to Arima Onsen (12 min, ¥980/1770 one-way/return). The ''Rokko Arima Katamichi J?shaken'' (_Z_b_L_n__________) combination ticket (¥1700 one way) works out marginally cheaper, and also lets you ride around the top of Mt. Rokko. This route is more scenic, but best avoided with heavy luggage.
It's quite easy to get around town on foot, though the uneven terrain may prove tiring to some. The JR train station has a large map of the area, but please not that it is not to scale and is also not oriented to the north.
Arima Onsen being one of Japan's famous hot spring towns, it goes without saying that one should pay a visit to one of the many baths. There are three public baths and a number of private ones.
Arima Onsen's website provides a list of baths and their contact information. They also provide information on pricing and accessibility.
[http://www.arima-onsen.com/eng/spa.html List of baths in Arima Onsen]
For a list of local inns and hotels, please refer to the page linked below.
[http://www.arima-onsen.com/eng/hotels.html List of hotels and inns in Arima Onsen]
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