Can I shoot modern ammo in it?
Short answer: Yes. BUT: Woodsmans made prior to 1933 were designed for standard velocity .22 LR. Those made after 1933 were all designed for high velocity .22 LR, with a stronger recoil spring and a case hardened mainspring housing, which is the part that takes the brunt of the recoil. The transition took place in the early part of 1933. They all will handle standard velocity ammo, which is what all target .22 LR is to this day. I do not recommend firing high velocity ammo in one of the early guns (before 1933). Just buy target ammo and use that. It is more accurate, and less noisy besides.
If your Woodsman is pre-WWII it will have a pattern on the mainspring housing, where the web between the thumb and forefinger touches when holding the pistol in firing position. If that pattern is checkered (left), it was made for standard velocity ammo. If the pattern is horizontal parallel lines (right), it was designed for high velocity ammo.
If it is a post WWII gun it will have no such pattern, because all were designed for high velocity ammo.