Can my opponent look at the cards in my archive at any time? If there's an agenda in my archives, and he wants to make a run, can he look at what cards are in there first?

All cards that are face-up in archives are open knowledge. Face-down cards in archives are not.

To expand on that, if your opponent wants to look at your Archives, hand him the face-up cards and keept he face-down cards to yourself.