Hi MonAleGut,
The short answer is yes, it should be possible.
When you plug in your Apple device to your new computer that's synced with your old computer, you probably get the "erase and sync" message.
Library Tunes will discover your old computer's iTunes Library ID (from your Apple Device) and apply this ID to your new computer's library. By doing this, you should be able to put new media files on your Apple device using your new computer. You should also be able to transfer purchases from your Apple device to your new computer's iTunes library. For non-purchases, you can use a program like SharePod to transfer.
Library Tunes will create a backup of your new computer's library just in case. There's a chance there's nothing in your new computer's iTunes library yet, so nothing to lose.
Again, I have to mention by advocating the use of Library Tunes, although it is a free program, I may receive compensation.