I did a little test in ye ole world builder. I placed a CG1 Bowman in a city with 20% cultural defense. Pretty much what you can have in any city. The Bowman was then allowed to fortify for 5 turns. Then I checked the odds for attacking. I did this with three decently promoted Swordsmen. Then I checked the the odds vs a regular archer. Then I checked all of them with walls added.
CR2 Swordsman vs CG1Bowman: 6 vs 6.30 28.8%
CG1 Archer :6 vs 4.8 70%
With walls CG1 Bowman 6 vs 7.2 21.9%
CG1 Archer 6 vs 5.7 59.5%
C1 CR2 Swordsman vs CG1Bowman: 6.6 vs 6.3 59.2%
vs CG1 Archer 6.6 vs 4.8 75.1%
With Walls vs CG1 Bowman 6.6 vs 7.2 26.8
vs CG1 Archer 6.6 vs 5.7 65.4%
C1 CR3 Swordsman vs CG1 Bowman 6.6 vs 5.4 68.8%
vs CG1 Archer 6.6 vs 3.9 95.1%
With Walls vs CG1 Bowman 6.6 vs 6.3 59.2%
vsCG1 Archer 6.6 vs 4.8 75.1%
Against mounted units they are the same as archers, but vs melee you should be pretty safe even without walls. I included the Combat1 promotion in case of an Aggressive leader. Shock axes definately stronger than both Bowman and Archers but cost 52
vs 37
So you can build 3 Bowman for every 2 axes. Availabilty of copper should decide your choice. As well as overall production. If copper is unavailable within the capital or 2nd city then Bowman are perfectly fine until Cats appear on the scene. Though getting metal is always a good idea.
Especially when you consider I gave the attacking Swordsmen a large advantage when considering promotions. Early on it will be rare to face CG1 CR3 swords. Most likely you will be facing CR1 or CR1C1 swords and axes.
CR2 Swordsman vs CG1Bowman: 6 vs 6.30 28.8%
CG1 Archer :6 vs 4.8 70%
With walls CG1 Bowman 6 vs 7.2 21.9%
CG1 Archer 6 vs 5.7 59.5%
C1 CR2 Swordsman vs CG1Bowman: 6.6 vs 6.3 59.2%
vs CG1 Archer 6.6 vs 4.8 75.1%
With Walls vs CG1 Bowman 6.6 vs 7.2 26.8
vs CG1 Archer 6.6 vs 5.7 65.4%
C1 CR3 Swordsman vs CG1 Bowman 6.6 vs 5.4 68.8%
vs CG1 Archer 6.6 vs 3.9 95.1%
With Walls vs CG1 Bowman 6.6 vs 6.3 59.2%
vsCG1 Archer 6.6 vs 4.8 75.1%
Against mounted units they are the same as archers, but vs melee you should be pretty safe even without walls. I included the Combat1 promotion in case of an Aggressive leader. Shock axes definately stronger than both Bowman and Archers but cost 52


Especially when you consider I gave the attacking Swordsmen a large advantage when considering promotions. Early on it will be rare to face CG1 CR3 swords. Most likely you will be facing CR1 or CR1C1 swords and axes.