how do I learned math in 1½ months

I'm taking an exam in a month and a half. Here is an old copy: http://www.studyjapan.go.jp/en/toj/pdf/013e.pdf

This is a big problem because I didn't take Algebra II in high school :( How would I go about learning this kind of stuff on the internet in a month and a half (or what are my best chances)?

Indeed... The rest of the questions are pretty standard fare, you just need to read through lots of things. If you don't get every question right you'll at least get some. Try tutorial.math.lamar.edu/AllBrowsers/1314/1314.asp for a bunch of free maths notes.