So I got confused by this too when I was prepping, so let me clear it up.
The "unofficial" score is the one you see on screen immediately at the end of your test. That one shows up right away, no waiting. The GMAT Focus Edition gives you your total score and section scores on the testing screen before you leave the test center.
The official score report (the PDF in your mba.com account) usually takes around 5-7 business days to show up, though it can sometimes be faster. It's not 3-5 days consistently, more like 5-7 in my experience and from what most people on this forum report.
For your specific situation with the school: if they said you can upload a screenshot, the cleanest thing to do is take a screenshot of the score on the testing screen right after you finish. Most GMAT Focus test centers allow this since you're doing it on the test center's own system display. I'd double check with your specific test center beforehand just to be safe, since policies vary slightly.
Once the official PDF lands in your mba.com account (under "Send Scores" or "My Account"), you can download it as a PDF and upload that. That's the cleanest document for schools. Most admissions teams prefer the official PDF over a screenshot anyway.
Short version: screenshot is available immediately, official PDF takes roughly a week.