I have a 4 month and a half month old myself.
Every single person's advice will differ, so you can use people's advice as a guide but a lot of it you will figure out for yourselves, and what works best for you.
I will share my thoughts on it however. We spent 4 nights in hospital before the birth and a night after, my fiance was getting a couple of hours sleep a night while in there (screaming women and various monitoring day and night), I slept on the floor with her each night, so was pretty shattered myself, but she was having to deal with so much more. Our little one would also feed up to 15 times a day, so we were getting a couple of hours sleep in 24 regularly, I would wake up each time she woke up anyway. When my little one was fed, I would try and get my fiance to go to bed for a little bit, even if it was only for 45 minutes, just to get any kind of sleep. Our little one was also pooping 12 or so times a day. After 6 or so weeks with so many problems with breastfeeding, my fiance produced too much fore milk, causing our little ones bum to literally get burnt by the acidic poops (literally looked like someone had run a cheese grater between her cheeks) causing her to be in agony everyone she went to the loo (bearing in mind how much she was going) and after trying literally every "trick" to try and rectify the fore milk issue, we tried formula, within 2 days we had a different baby. It broke my fiance's heart because she wanted to breast feed and felt like she failed our child by causing her the pain, but she did an incredible job by doing everything she could. I personally do not agree with mothers that choose not to breast feed just because they don't want saggy boobs, or the night feeds etc. Baby's are hard, but do what is best for them. So it is great your partner is doing that. Expressing is really, really helpful for you both when she gets the hang of it (took my fiance a little while to get it figured out) too.
Regarding getting up at night to help out even while at work, yes, I do think you should get up and help when you can, I often got by on only 3 or 4 hours sleep in 24 and worked full time. The mother will be more tired and more stressed, I can almost guarantee it. It won't last forever, keep telling yourself that!
Also, like many have said, there will be arguments, and jabs at you etc, just let them go, you are both pushed to your limits right now, again, it won't last forever.
Best of luck with it all!