Hi, I know this scenario well. My oldest is now 2 1/2 years old and my youngest is 7 months. When my son was 0-3 months old he sometimes nursed up to 15 times per day and I often did everything on my own b/c my husband travels for work. It was a rough go for us too in the beginning, lots of acting out and jealously from older son and guilt on my part (how to spend quality time with both). Even now, we are constantly readjusting to our kids' needs.
My son climbed all over me everytime I nursed too. Tough one to deal with because they are used to being your baby and its so hard for them to understand their role, the new baby's needs etc... We tried so many things. First off be firm about setting limits - this is how we behave when mommy nurses, explain in simple terms why you nurse (baby needs to eat), provide an outlet of activity for your older child and praise positive behavior (be absolutely silly about this). We had a special box of things for my son to play with so that everytime I nursed, the box came out and was put away after each nursing session. He also had a snack during nursing sessions, played playdough, watched a movie or PBS in his booster chair (safe but somewhat confined area for play/eating). We used time out for the big stuff and eventually for staying within the limits (no climbing on mommy, baby etc..). First off it was a total wash b/c he was so young but after a month or two the response and use of time out when needed began to click with him and his behavior improved. Lots of verbal praise and a few stickers here and there for very specific behavior.
The biggest thing that worked for us was keeping our oldest son's schedule as much as possible. Really sticking to wake up time, naptime, meal times and bedtime were huge. Its very hard when you are on your own to do all of this but maintaining the schedule helps your son feel secure and I found that it helped me get through the day even with all the nursing times. I also had to make a huge point of finding time for my son without the baby around. Whenever my little one slept, I made a big deal about my oldest son.
I'm sure you are like me, trying to find time to do just the every day things can be daunting not to mention find quality time to spend with both kids. I used my baby pack/carrier a lot to keep my little one close and still be able to accomplish things around the house and with my older son. I also found involving my older son in household tasks was not only a great way to interact with him but a huge help. When we did laundry he helped sort, dishes he helped take out of the dishwasher and he helped make meals with me too. These big boys tasks really made him aware of his role in our family and helped him feel important. Not to mention I was able to do the basics without worry of what he was getting into or taking away from time we could spend together.
Bedtime is a challenge. If you are able I recommend the baby pack. I was able to give my oldest a bath while holding my youngest in the pack. When it came time for stories and cuddling before bed, I put our little one in the bouncy seat or swing while I gave my full attention to our oldest. I had to keep bedtime routine brief often to attend nursing or a fussing baby but there was huge difference when I gave my older son even 5 full minutes of my full attention at bedtime.
Last but not least, don't heap on the guilt. This mommy gig takes a lot of finesse and on the job training. Having two kids is so different than just one and when they are close in age they need so much attention. I wish you well and hope it helps to know there other moms know your situation well and are making it through each day too. Take care!