I'm planning on using my laptop in Japan. I have a Type H power adapter:
Is there another adapter I can use? Or should I but another cable when I'm there?
Type H seems to be a relatively rare type of pin as your link specifies; only used in Israel and Palestine. I assume most shops won't carry an adapter for this kind of plug.
That said, the other end is a IEC 60320 Type C5 as user CSM mentioned, and those are readily available in stores in Japan at least. Depending on where you will be, you can go to a Yodobashi Camera store or BIC Camera (Don't worry about the Camera monikers, they have all sorts of electronics) and show them a picture or take your power charger and show them that you need a cable for the end you plug in.
Will probably set you back around 500-1000 Japanese Yen (5 to 10 USD give or take).