There are a multitude of different ways to say "got some shuteye". The phrase itself is an informal way to describe getting some sleep or taking a nap. Some synonyms for "got some shuteye" might include "caught some Z's", "hit the hay", "took a snooze", "dozed off", "slept soundly", or "caught some rest". Each of these phrases is slightly different in connotation and context, but they all refer to the same basic idea of getting some needed sleep or rest. Whether you prefer to say "slept soundly" or "caught some Z's", all of these synonyms help to convey the idea of taking a necessary break from the waking world to recharge and rejuvenate.