Great Britain star Proud finished 23rd overall out of the 115 competitors, with only the top 16 progressing.
The 20-year-old, who reached the final of the 50m butterfly, clocked a time of 49.35 seconds.
Sadauskas, competing of Lithuania, was ranked 27th with a time of 49.49.
Former Plymouth College and Leander swimmer Calum Jarvis, Great Britain’s other competitor in the two-length event, also missed out, finishing 32nd in 49.68.
Proud will swim in the 50m freestyle competition on Friday – an event where he is British record holder and Commonwealth Games champion.