If I use Golos, Tireless Pilgrim's activated ability and hit a card with cycling, can I cycle it? And if so I assume I can't do so for free?
No, you can't cycle cards exiled with Golos.
As Golos says, you exile the top three cards of your library and then you may play them without paying their mana costs. Cycling requires the card to be in your hand rather than anywhere else, and "exile" is its own zone.
702.28a Cycling is an activated ability that functions only while the card with cycling is in a player’s hand. “Cycling [cost]” means “[Cost], Discard this card: Draw a card.”
Also, "playing" a card, as Golos allows you to, means casting it as a spell or playing it as a land, whichever fits:
701.13b To play a card means to play that card as a land or to cast that card as a spell, whichever is appropriate.
As you correctly noted, cycling a card means activating its cycling ability, not playing it as a spell, so you would have to pay the cycling costs even if you could play the card from your hand without paying its mana cost.