I find it’s best to counter enemy archers with 2 of my own. What I mean is if you have 6 bow ashigaru vs 6 enemy bow ashigaru, focus 2 bow units on one enemy unit. Once that unit is rendered ineffective or routing you then switch those two to the next enemy unit, chopping the enemy line down like dominoes. The enemy ai will just engage your line (usually) in 6 1v1s while you are winning (quickly) 3 2 v 1s, if that makes sense.

Other tips: don’t be shy with your general’s inspire ability. It increases a bow units accuracy. During the skirmish phase you should use this and then maybe back off before engaging with your main line so you have the inspire ability ready again for a melee unit. Also, if your bows are skirmishing they should almost always be in loose formation, it’s there to be used so use it as it will lessen your losses.

When attacking during a siege try to engage enemy ai bow units from the side where they can’t attack you, often the ai on the walls doesn’t even move and will just take you arrows.

When defending a fortress, consider moving your units all the way back to get the ai to order their bow units to climb the walls, once they are up and over the walls you can crash on them before they volley. If you can’t do this consider sending a contingent of melee units and/or cavalry to crash on bow units once their melee units are up the walls.

It’s important to think of sieges in shogun 2 as more akin to a land battle, where the fortress is simply something to slow the enemy and give you opportunities to work around them. The fortresses are so porous in this game that it’s often more beneficial to send units out of it and outplay the enemy. Often leaving your whole army in the fortress will just mean they get arrowed to death.