This booklet you want to obtain is going to be 'double'? I mean, should the text be on two sides of a given page (like a regular book), or should the text be as you wrote, i.e. one-sided (that's how I understood)?
First, install a virtual pdf printer program (eg PDFCreator). This will allow you to prepare a PDF for the printing house.
If it's going to be a regular booklet (double-sided) then just turn on booklet printing (page scaling) in Adobe Reader, select a subset with 'on both sides'. Set PDFCreator as the printer and print. There should be a PDF ready for printing. Take the resulting file to the university and tell them to print on both sides (if possible). After the appropriate folding of the printed sheets, a booklet will be created :)
If it is to be one-sided printing (as you wrote 1 of 20, 2 of 19 etc.) then the only solution that comes to mind is to create 10 PDFs with two pages each. You do it so that you select page scaling to 'many pages per sheet' and as the print range you set in turn 1.20 then 2.19 etc. You print to pdf using PDFCreator and take the prepared files to the university.
This operation is not very complicated, so you should be able to handle it.
Greetings.
EDITION
If you want the second option (single-sided printing) then I've just noticed that the sequential feeding of the pages for printing also works. So you don't need to create 10 files. You simply enter 1, 20, 2, 19, 3, 18 as a page range ... and the rest you do as described