M.J.
Yes, it does get easier. I hate to tell you this, but you need to suck your guilt up. Sleep when the baby sleeps is NOT stupid advice, it's tried & true advice which is why it's been given by so many mothers for so many years. If your Mom is wonderful enough to take the 4 year old for a day, drop the guilt like a hot potato & take a freaking nap already! If your husband comes home from work & holds the baby and makes dinner, drop the guilt & take another nap if you need it. Before you know it, you'll start to feel like a human being again & not the walking dead & you'll be a much better Mama all around. If you're sleep deprived for an extended period of time & horrible to yourself the rest of the time, well, what else can you expect but to be miserable, right? Take care of you, then you'll find that getting up in the night with the baby won't be that big of a deal & dealing with a 4 year old who's rougher than she should be will be easier to handle as well.
*Edit*
I forgot to mention when I first posted that my kids are only 17 months apart, so I have personal experience with this, too. My oldest was still in diapers & sippy cups when I brought a newborn home after having a c-section & couldn't even pick him up to change his diaper. When the baby was 6 months old we had a total of 40 hours to prepare for my husband to deploy for 4 months while we were in the middle of apartment hunting. If I could get through that at the age of 24, you can get through this, I promise, promise, promise!!!