The famicase site provides most of the information you need. You probably should read the full submission guide from super meteor. If you can't read japanese, here is a link to google translate (Which you may also want to use to generate japanese translations for your cart text. It's only polite. It's the least you could do. You could probably do more.)
The submission instructions specify 90mm x 43mm dimensions with 300dpi. If you've only worked in digital media, that wont make sense to you. As a digital being, I have never seen or touched a "mm" or an "inch" and I'm not really sure what they are. All I can say is that when I create something of 90x43 "mm" in inkscape and export it at 300 "dpi" I get a resolution of 1063 x 508 pixels. I have made submissions in this size before and they were accepted so it must be okay.
Likewise, your logo should be 591 x 591 pixels, and it should be solid black on white (although it seems from the samples that anti-aliasing is allowed)
Once you have your correctly sized pngs, use imagemagick to convert them to CMYK jpgs
I set the quality to 97 to make sure there aren't any obvious jpg artifacts. You might like to go all the way up to 100. It never ended up being bigger than a MB when I tried that.
convert famicase_case2019.png -format jpg -quality 97 -colorspace CMYK label.jpg
Similarly, for your logo
convert famicase_org_logo2019.png -format jpg -quality 97 -colorspace CMYK logo.jpg
If you run `identify -format '%[colorspace]' case.jpg` it will now say "CMYK". (This is the only way I could confirm that the conversion was doing anything at all. Colorspace doesn't really seem to mean anything to any digital entity other than printers)
Now, fill out this form (it's from the samples)
1.お名前またはペンネーム <your entry name>： (we must assume that this field is asking for your name, as they do want that information and none of the other fields ask for a name, and the only other way I'd interpret "your entry name" clashes with field 3) 2.職業 <your job>： 3.タイトル <title of your game>： 4.ソフト会社 <name of your game company>： 5.カセットカラー <cassette color you want>： No.1: No.2: No.3: 6.カセットの状態 <cassette condition you want> good or dirty , or anything is OK 7.解説250文字 <story of your game(250 letters)>：
Then send that to email@example.com along with your case and logo images.