First, I’ll just transcribe the text in that picture:
おしろへ あそびに
きてください。
ケーキを つくって
まってます。
ーピーチよりー
Translated, it would be:
Please come to my castle to play.
While making a cake, I’ll be waiting.
ーFrom Peachー
As for the syllabaries used in the Japanese version, it’s all hiragana and katakana in that picture; no kanji. Everything in the picture is in hiragana except for ケーキ (cake) and ピーチ (peach), which are in katakana. You’ll tend to find most older games are like this (with no kanji), usually because of a combination of technical limitations (there are a lot of kanji out there!) and making it easier for young children to read. However, Japanese usually has no spaces in its sentences (except for the built-in space in punctuation like quotation marks, commas, and periods). For text with only hiragana and katakana in it, this would be extremely difficult to read, which is why they usually add a space after particles in such text (like in your picture above).
To illustrate, this is how your above picture would normally look in Japanese (written with kanji):
お城へ遊びにきてください。ケーキを作って待ってます。
While it would look like this written with only kana:
おしろへあそびにきてください。ケーキをつくってまってます。
With the all kana version, it’s a lot harder to see where words begin and end than with the kanji version.
Anyways, hope this helps!