I would classify lassitude as common and alit as rare, TRex, because lassitude is used more often!
It's an alternative to alighted, as the past tense of alight, meaning to get out of a vehicle or to find by chance. We class alighted as common, but not alit. The Corpus of Contemporary American English has 26 instances of alit, compared with 159 for lassitude and 162 for alighted.
Alit is even rarer in the British National Corpus: only one example (from a poem by Ted Hughes, "And the dove alit / In the body of thorns"). This corpus has 88 examples of alighted and 32 of lassitude. I found similar results in Google Book Search and various newspaper archives.