Do I need to know Japanese in order to play Super Princess Peach?

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!).

In theory no, in practice yes.

You are right that Super Princess Peach is not a story-heavy RPG with long reams of text that tell you what city you need to go to or whom you are expected to confront next. If you have played a platformer before, the basic controls should be fairly intuitive and easy to figure out.

Where things start to fall apart is that the game (primarily through Peach's intelligent parasol) gives you hints here and there as to how to solve various puzzles you encounter in the game. I have the USA version in English, but the game appears to be the same across all regions. The puzzles often have to do with some interesting ways to use your Vibe powers or move in just the right way to avoid falling off some obstacle. I found that some of these puzzles can be figured out with a general roleplaying mindset of thinking about the game environment and what might logically work if the game were set in real life, but a few of them were downright non-intuitive and I probably would not have figured them out without hours of expended emotional energy (both my own and Peach's).

Many (but not all) of these puzzles are not really required to get through the main game, but are necessary if you are striving for 100% completion (rescuing every Toad, finding every puzzle piece, etc.).

There is also an in-game shop where you can purchase power-ups with the coins you have gathered. You select your desired item in the menu, so you will need to be able to understand what it is you are purchasing. If you are willing to forego these power-ups or simply purchase everything in the catalog this might not be a huge issue, but if you are short on coins and need to make every one count this could be an issue.