Does Cycling a card to trigger an ability allow you to target a creature protected by the color of the card you're cycling?
No, the angel with protection from Blue cannot be targeted by the ability of Choking Tethers.

The relevant rule about Protection:


  702.16b A permanent or player with protection can’t be targeted by spells with the stated quality and can’t be targeted by abilities from a source with the stated quality.


In your case, the source of the ability, Choking Tethers, has the quality 'blue'.

Note that (as mentioned in the card's Gatherer rulings) it is allowed to cycle Choking Tethers even if there are no legal targets for the triggered ability (e.g. there are no creatures at all, or only creatures with protection from blue). This is because the Cycling activated ability itself does not require any targets to activate. The triggered ability that taps a creature will not successfully be put on the stack, but the Cycling ability will resolve just fine and you will draw a card.