The card has 32 bytes of plain text which contains some header information and my student ID then there's 48 bytes of encrypted data for a total of 80 bytes. It appears I can write 144 bytes to the ring.
I've found out that the card uses the ISO 14443-4 specification which the ring does not support, even though it has enough storage space. The secured area requires a hardware DES chip so that's completely out of the question. Getting the reader at school to even read the plain text (in any case it probably also needs the encrypted data) is probably not going to happen even, although if I can find a spec for the data format I'll try it. The reader probably won't read any other type of chip anyway and its not possible to change the chip it reads as.

I learned quite a bit about the chip in the ring while researching this at least.

If I can convince my school to do it, would it be possible for them to write the same data on my student ID to my ring? The student ID is a Mifare EV1 card. I understand that this is probably not possible, it would just be so cool to be able to open my room at school with it.