Since you have access to the old computer, with its iTunes library intact, transfer the complete iTunes folder from the old computer to the new computer. You can use this procedure to backup your iTunes folder to an external drive
Back up your iTunes library by copying it to an external drive - Apple Support
To summarize, the first step is to "consolidate" all of your iTunes media files so that they are in the iTunes Media folder in the iTunes folder. The second step is to copy that iTunes folder to an external drive. The third step is to "restore" that backed up iTunes folder on your new computer, to replace its existing iTunes folder. Since the new computer's iTunes folder has new content (that's not in the old computer's iTunes folder), with iTunes not running, move it to a different location and put the iTunes folder from the old computer in its place. When you run iTunes, it uses that iTunes folder from old computer. Then, add the new content from the other iTunes folder (that you moved), OR if all the new content is from the iTunes Store, you can just download it again from iTunes Store Purchased screen.
Download your past purchases - Apple Support
NOTE: If you know how to set up file sharing between the two computer, you can transfer the iTunes folder directly, instead of using an external drive. Depending on network speed, using an external drive may be faster.
