Were Alī ibn Abī Ṭālib and Mu'awiya I alive at the same time?
Yes. Alī ibn Abī Ṭālib and Mu'awiya I were alive at the same time for 57 or 58 years, from AD 603 to AD 661.
Alī ibn Abī Ṭālib was 3 or 4 when Mu'awiya I was born.
Mu'awiya I was 57 or 58 when Alī ibn Abī Ṭālib died, aged 61 or 62.
Alī ibn Abī Ṭālib
4th Rashidun Caliph (r. 656–661) and first Shia Imam
Born 17 March AD 599 · died AD 661, aged 61 or 62