There are several ways to look at your question. One is that muscle soreness is a good thing. You can't build muscle without soreness. DOMS results from microtears in muscle tissue. Repairing those tears creates a stonger muscle. Excessive DOMS which prevents one from moving about easily is another matter and means you are working too hard. You should look into how you can rebuild that tissue more quickly.

Upper body weight lifting does pretty much nothing or is contraindicated to improve climbing.

If you have to walk your bike up hills, you need to improve some aspect of your strength or conditioning. So the question is: Why do you have to walk your bike? You can't turn the cranks or you run out of breath? If you run out of breath, you just need to ride more hills. If you can't turn the cranks, either your legs can't produce the necessary force, or your legs get tired too easily. If your legs get tired too easily, you just need to ride more hills. If you can't produce the necessary force, you need lower gears.

Weight training can help, but riding your bike up hills helps more. Another question is: How's your motivation? How much pain can you tolerate in pursuit of your goal? I have walked our tandem a couple times when we were touring loaded and encountered hills of over 20%. I have never walked my single, even when encountering grades of over 20% over 200 miles into a ride. You just have to gut it out. One becomes stronger through the principle of overload. You have to do more than you think you are capable of. Make yourself do it. Again and again. When you are sore, ride anyway, but ride very easily. Don't ride hard again until the soreness has passed. You won't believe how quickly you'll get stronger.

A very simple drill to make your legs stronger is one-legged pedaling. Prop one foot in the frame triangle and pedal with the other foot until it won't anymore. Force yourself to keep a tight chain. Switch feet and repeat. Then pedal with both legs for a little bit. Repeat until exhausted. Ride home easily. This is best done on a short ride of an hour or less, so you don't have to bring water bottles.