Archers are certainly dominant early in the game, as they should be. Well, perhaps dominant is the wrong word, but they require careful consideration to defeat them. Remember, before catapults, you are attacking into that massive cultural defense bonus, added to their 25% city defense bonus to start.
50% Culture, 25% to start with, and CD 1 (20% City defense) garners
3.00 x 1.95 =5.85
1 First Strike
And an axeman with CR1
5.00 x 1.20 = 6.00
Now, correct me if this is a bad estimate, but that gives the archer about 51%-54% to start with, and with every one it fends off, it only gets more experienced/ stronger. And with a barracks, (even if you have no other ways to garner experence early), they will be eligible for another promotion after beating one axeman.
Simulated Battle #2
Archer defeats Axeman 1 (~40% Health)
Archer promotes CD2 (~70% Health)
Archer rests one turn (~80-85% Health)
Archer
3.00 x 0.80 x 2.15 = 5.15
1 First Strike
3.00 x 0.85 x 2.15 = 5.47
Axeman
5.00 x 1.20 = 6.00
Still not very much of an underdog.. 35-45%?
Edit: Added Thoughts
Keep in mind, that with a 25% terrain bonus, those numbers become 5.76 and 6.12 (check my math on those last two). Perhaps, with first strike, from about 50/50 to a favorite on the 2nd battle.
And this is only giving the archer one turn of rest.