In contrast to “arch” /ɑːtʃ/ (aatch) (UK), /ɑːrtʃ/ (artch) (US), “archangel” is pronounced with a hard “ch”, i.e. /ˈɑːkeɪndʒəl/ (aak-eyn-dzhəl) (UK), /ˈɑːrkeɪndʒəl/ (ark-eyn-dzhəl) (US). You can remember that the Mormon Church teaches that Noah (the builder of Noah’s Ark) is actually the same person as Archangel Gabriel, if it helps you to remember the correct pronunciation.
An archenemy of God would be the “opposite” of archangel, which will hopefully help you remember that it is not pronounced the same; it is /ˈɑːtʃˈɛnəmi/ (aatch-en-ə-mee) in the UK and /ˈɑrtʃˈɛnəmi/ (artch-en-ə-mee) in the US.