Don’t worry, you’ll be fine. Get the new passport and apply for a visa. Since you want to study for a year, make sure that’s what the acceptance letter says (instead of just 3 months). Make sure you bring the letter of acceptance with you to Taiwan. You can always bring the expired passport as well, but that shouldn’t be necessary.
There is no student visa as such. There’s a visitor visa (with an FR endorsement), and a resident visa. There are some advantages to coming on a resident visa, so try to make the TECO (“embassy”) issue you with one. Coming here for a year, that’s what you’re entitled to anyway, but it’s always at the discretion of the local mandarin.
If for some reason all they can offer you is a visitor visa then it might not be worth it to apply for it, considering that as a German national you’re now allowed to stay here visa-free for up to 90 days (then leave the country, come back even the same day, and the clock starts anew), while the visa will be valid for 180 days and might set you back around €80. Save yourself the hassle and instead do a trip abroad every now and then (which is what most foreign students here do anyway).
Apply for the visa with the letter of acceptance, scholarship certificate, and perhaps the proof of sufficient funds (bank statement) but that’s optional. What kind of scholarship do you have? If it’s the Taiwan Scholarship or Huayu Enrichment Scholarship, the visa should be free of charge.