Your bow is faster than her ranged options. Palu is likely to feel the burden of having to approach as long as your bow usage isn't predictable. She still has a reflector, so you can't telegraph your bow or she will just send it back.
With bombs and boomerangs, you need a different approach here. Instead of targeting her directly, you should be using your bombs and boomerang to limit her choices for movement. Palu is very comfortable in the air, so Z-Drop your bombs to deny safe landings, aim the boomerangs high to discourage jumping and leverage the returning rang to limit her options so that she can't move comfortably. This makes it easier for you to predict where she goes so that you can counter air movement with your sword. Young Link has the advantage of a disjoint in the air, so you should be able to use FAirs to contest her air game. She will have trouble landing against you, since you have that disjointed UAir that practically forces her to Warp/Airdodge to get back to the ground. Your bombs and boomerang can limit the places where she can safely land as well.
If you use your projectiles and aerials well, she will be forced to a ground game. Nothing is straight forward on the ground for either character; her tilts have a lot of range and her dash attack is good. Your sword isn't as oppressive here. However, her tilts are not particularly fast, so your job is to be up in her face and make her feel pressure. This way, she won't be as comfortable with sending out a tilt. If you let her get too comfortable, she sit outside your sword range to throw out her best attacks, but not far enough to let you pull bombs or rangs safely. Palu will be looking for jabs, grabs and dash attacks to get you off of her. However, your grabs, SH Fairs and dash attacks have a lot of range, so use that to your advantage.
Essentially, try to be far away so can play your projectile game, or be so close that she can't play her keep away game. The middle range is her domain in this matchup, so limit her ability to be in that comfort zone.