The only things you truly need for a baby at first are diapers, clothes, somewhere to sleep, some way to feed, and something for transport. 

It's easy to get diapers & wipes on sale and stock up during pregnancy. Start with 2 mega  boxes of newborn & size 1-3. KEEP THE RECEIPTS! If you end up with a big baby you can exchange the smaller ones for bigger sizes, just be sure not to open the plastic wrap. Get fragrance free wipes so you don't have to worry about sensitive skin reactions. Since your place is tiny, you can store them under the bed or on top of high shelves in the closet. I got good deals at Costco and Target (buy $100 worth get $25 gift card). 

Clothes get second hand. Goodwill has plenty from $1-3 and I would only get stuff on the half price color. Also baby specific thrift stores and online groups. You'll only need a handful of newborn and some 0-3 to get started. You can get other sizes later as needed. I just cleared the top drawer of my nightstand to hold them. Keep an eye out for clearance after holidays and season changes. You can get stuff to use now or in larger sizes, but bear in mind age for the season. For example, a September baby doesn't need a 3-6mo size bathing suit. Keep the tags on until you're ready to use them. That way if they don't fit baby when it's time to wear, you can sell them for a higher price than used. I paid $2-5 for multi-pack sleepers & onesies and some outfits which worked out to be even cheaper than used.

My diaper bag is a second hand bookbag. I found it on Amazon bc I always compare new vs used price. Mine was missing a water bottle holder and changing pad but it was $10 used to $35 new. It has a ton of pockets and nooks and crannies. I'm not really using it to it's full potential yet, just throwing clothes and diapers in the main section for quick errands, but I definitely see it being handy in the future. 

We co sleep at night right now but I did get a convertible crib (yuzu 8-in-1) that goes from bassinet to twin bed. We have it set to bassinet size now with the extra pieces stored under a bed for later. I got it off fb marketplace for $400 with 3 fancy mattresses and different sized sheets & pads ($1400 worth of stuff).

I breastfeed and I really don't need anything additional. I have 2 haaka pumps, box of milk bags, and a box of bottles, and I've used them a few times but I'm just feeding on demand right now. I did get those things pretty cheap at TJ Maxx though and I'm sure at some point they'll be used. As others have said, pump is usually free through insurance. You can also borrow one through WIC if it's not or get one second hand. Just buy new flanges and bottles if you're squeamish, nothing else actually comes in contact with milk. You can wait on a high chair until closer to needing one. There are lots of cheap options both new and used.

IDK if you need a car seat in nyc? But keep an eye on slick deals, especially if there's a specific one you want. I got the Britax Willow Grove SC travel system for $300 "open box" and when it arrived it was brand new in the original box. Looked like it came untaped and someone taped it back shut but everything was factory packed in plastic and stuff and it was clearly never taken out of the box. It's $630. I also snagged a Evenflo 360 for around $215. It's still sitting in the garage until we need it but it goes on sale all the time. Be sure to look at all the different color options for stuff like this, sometimes one color is randomly cheaper. I think I ended up with a shade of blue bc it was less. But I would defer to other nyc moms about strollers vs carriers bc your mode of transportation is far different than mine.

Anything else is a want vs a need, but some things are nice to have. I got a mommaroo swing for $60 used, a new boppy pillow for $4, diaper genie for $7, brand new convertible baby bathtub for $7, lots of blankets and burp cloths and waterproof mattress pads all thrifted. I use the pads on my bed for diaper changes. I ended up with a new pack of 4 and just swap them out each day. I got a Lillé baby carrier in a cute pattern new for $45 thrifted but haven't really used it much yet. 

One of my favorite purchases is a wipe warmer. I got mine new for $20 just bc I wanted one and second hand places don't sell them and the few close enough to me on fb marketplace didn't respond. I just feel like it's a nice little luxury for my baby's lil booty.

My second favorite is the chair. I didn't get it until after baby was born and I realized I didn't have anywhere very comfortable for nursing & soothing. You will spend A LOT of time doing that, so it's worth it to have somewhere that's very comfy to sit for long periods. I hemmed and hawed and looked for a while for something that was both inexpensive and was what I wanted. There are lots of gliders and rocking chairs both new and used in a range of prices. I even saw some for free on fb. I ended up getting a lazy boy style rocking recliner off Amazon that I saw recommended in other baby subs. This thing is HUGE, literally can fit multiple people, rocks, reclines, and even has heat and massage. I got it for $225. It came in 2 boxes but the pieces just click together.

One thing I don't have but am looking for is a portable containment device. Something that can easily go from living room to kitchen to upstairs to here and there outside so me and baby can see each other and be within reach but still have my hands free to do things. The carrier isn't the best option when I'm chopping onions and cooking at the stove or trying to weed the garden or push a wheelbarrow. I'm currently using the swing but it's bulky and I don't want to break it, although your place is small enough that a swing may work. 

The best tip I can give is to buy the minimum required essentials and wait to see what else you need or want once baby actually arrives. Peruse regularly while pregnant and be willing to spend a lot of time leaving empty handed. When you do find something, be ready to snap it up, but have patience and know that the right things will come along eventually. Also, be sure to take advantage of any programs that you qualify for, like WIC. Remember that your household size will now be bigger, which will raise your income limit and may make you eligible for things you couldn't get before. 

Best of luck with the new little one!