Historically, the three most common regions were US, Europe and Japan. Each game (and the system) would have the respective language options for that region, and games would usually be locked to that region only.
Nowadays, most consoles, including the DS, are not region-locked anymore. Note that digital games might still be locked - such as DSi content. While you can most likely run the game, there might not be language options for your system's language. Simply put, the game cartridge only holds the JP language, just like a European cartridge will often have English, German, French, etc.
If there is not an automatic language switch, or an obvious way to change this in settings, there is very likely not an option to do so. In my experience, a game with multiple language options will prompt you for one on first launch.
This often has to do with different release dates, additional development time, or even exclusive versions to that region. For example, this specific game was released in Japan a few months prior to other regions.
To answer the primary question: typically, games like this (Platformers) are very light on story, if based on one at all. The major issues from language that will arise are instructions, tutorials, text hints and perhaps button prompts. Most of the time, these games should be simple enough to play through without knowing the written language. Even so, with internet and translation tools, it should not provide an insurmountable challenge. You might find some useful information on the Mario Wiki page, describing gameplay elements, extra's and the plot (mind the spoilers, though!).