Hi J. - YES, you can do it!!!
You sound like me, this time last year! I had number two on December 20th, number one turned 2 on January 7th.
My greatest fear was how was I going to handle going out with two, by myself. And the winter, breastfeeding etc.
It seemed to work itself out. First off, I did stay home more, play indoors more. Instead of meeting the BIG play group from my MOMS CLub, I just had one other Mom and her child/children over for play dates.
As far as venturing out. It does take some planning. I would get the baby ready and in his infant carrier/car seat. Then I would leave him in the kitchen, while I loaded up the older boy, diaper bag etc in the car. Then come back and load the baby in the car. At the grocery store, as much as I think those shopping carts with the cars built in are germy, I let my older son ride in it and he thought it was so special, kept the baby in the carrier and quickly got what I needed and left. I also recommend parking near the cart corrals whenever possible.
For breastfeeding, when in public, I either returned to my car and gave my older child a snack strapped into his seat. Or put the older child in the stroller with a snack while I fed the baby. That way, I knew he wouldn't be running off while I was exposed and I wouldn't need to chase after him.
As far as being outdoors in the cold, I am from S. California and have only been back here for a few years. I get a little freaked out in the really cold weather. So I had the snuggly that goes around the infant carrier to keep the cold wind off my baby. And my older child just stayed indoors for most of the winter to play instead of playing outside. But if we did have a nice day, I would bundle us up for a short 20 minutes outside. Just keep hats & blankets on the baby, cover those ears!
I did buy a double stroller and used it a lot last winter. By this summer, I was back to using my single stroller and my 2 year old learned to walk along side. He even walks beside the grocery cart now and doesn't run off. I also had a sling and a back-pack style carrier for hands-free carrying of the baby at times. These worked really well when out in public too.
As far as time alone with each child. The baby is on a regular 2x a day nap schedule now. So, I have alone time with my older son during the baby's morning nap. Then after lunch, I put my older son down for a nap 1st and have some alone time with the baby before he goes down for his nap. That leaves me about 45 minutes a day where both boys are sleeping and I can clean a toilet, start on dinner or do some thing for myself.
Recently, the two of them have started really playing together. It is so sweet to see them chasing each other across the floor, splashing in the bathtub together or just giggling together in the back seat of the car. I am happy to have two boys, close in age, because I think and hope that they will be best friends for life.
Congratulations on your new baby. You have some really special times ahead of you! It will be wonderful! Good luck