Something I didn't see a great deal of discussion in the corebook around is how to do extended contests. These are the sorts of things that the Skill Challenge system in D&D4e attempted to address, structuring non-combat conflicts in a rigorous way. This allows everyone to get involved and resolve such situations in a more dynamic way than "make a single check to succeed".
The specific sorts of contests I have in mind are athletic competitions - the Greek games such as Olympian, Pythian and others, as well as less official ones. Some events are easy - all the ones involving combat.
But then you've got things like races (including one in armour), accuracy throwing/shooting and distance throwing. A single roll to see who wins is distinctly unsatisfying, especially in the case of the races. I could see the target events being a roll for each shot/cast, would you then employ some sort of margin? Perhaps the highest successful roll is the best?
For the races, perhaps divide it up into start, some collection of intervals and last leg/finish, with a roll at each? Maybe some sort of cumulative bonus/penalty depending on previous sections?
I used to play a good deal of 4e and I was fairly comfortable with molding the skill check system into a model of just about anything I wanted to do. In ACKs, however, there are often fewer explicitly spelled out concepts, and yet the ones that are spelled out are fairly specific.
The end result is that I often have to start off with trying to imagine how things might “really” work and then try to deduce a model from there, possibly adding a die roll on top to keep things from being deterministic. Unfortunately, these ideas are often all completely different.
For a race, you could take two tacks. The first is to simply calculate someone’s average speed over a large number of rounds based on their movement speed (even in armor) and the maximum number of rounds they can run based on their constitution (described in the book). Another option might be to try and replace the skills used in wilderness/dungeon chases with ones more relevant to trying to pass each other. A few ideas might be applicable such as saving throws vs. paralysis for tight turns to avoid falling down (or perhaps just slowing down in that case).
For archery, if the contest is one shot then that’s really just modeled by the roll. If you wanted to make it more complicated you might think of additional factors and roll them into several modifiers to a 2d6. To elaborate: say the player is in an archery contest with one other person, you could give the player +1 if they have the greater attack throw, STR score, DEX score, WIS score, -1 if the opponent has a better, 0 if tied. Then roll 2d6 with the modifier. say 2 or less is a decisive loss, 3-5 is another shot (roll) at an additional -1 6-8 is another shot with the modifier “pulled” one towards zero, 9-11 is another shot with an additional +1, and 12 is a win. This can create an ebb and flow that you could narrate while (hopefully) eventually producing a winner or loser.
If all that sounds like a huge hassle and not worth it, why not just use the 4e system? there’s no skills per se, but you have stats for which one will always be relevant, and if a proficiency seems relevant you could ad-hoc a +2 or +4 bonus.