What part of speech is 到 (to) when used in 两只饿到眼绿的老鼠正在房梁上打架?
You are essentially correct. Here, 到 is a preposition that, depending on context, can represent "to," "(up) until," or "up to." 

e.g. 到目前为止 : up until this moment

工作到天亮了 : work until the sky becomes light

从周一到周四 : From Monday to Thursday

In your particular example, there seems to be an implied “的程度" at the end of the descriptor which is omitted, i.e. one can write 两只饿到眼绿的程度的老鼠正在房染上打架, and in order to prevent a sentence from being clunky, this has been omitted since it's clear from context what is being said here.