The answer really is to go fast. I say this as a long time UR player in Modern. By go fast, I mean you want to pressure your opponent's life total and build big creatures.
It looks like you have Master of Etherium in the main. That's a great card for you because it can often push your creatures out of Pyroclasm or Anger of the Gods range. UR decks are typically Snap/Bolt decks. They rely heavily on damage based removal, so if you can bump your creatures out of that range (~ 3-4 toughness), you'll be doing the most good. Etched Champion has protection from just about any removal UR has, aside from countering it. If you can land an Etched Champion you are in very good position.
Holding double black up to re-equip Cranial Plating at instead speed may be difficult, but it's a great way to get around targeted removal during the combat phase. They shoot your equipped Vault Skirge? Re-equip it onto your Ornithopter.
If you want to play around counterspells, be aware of what the opponent has already played that game (and round) and take note of their open mana. They have one blue open and haven't played Spell Snare all game? Don't play your 2-drop. You can also try waiting to play multiple spells per turn. Counterspells will typically cost more mana to play than your creatures will, so even if a creature or two of yours gets countered save your best for last to get it on the board.
Lastly, most blue decks (and UR especially) tend to board out counterspells against fast decks like affinity. They often bring things in like Shatterstorm, so don't over-commit to the board. Be weary of sweepers as well.
You are always the beatdown in this matchup, UR doesn't have Splinter Twin anymore, so as long as you can keep dropping threats you will likely come out on top.