Does this electric panel meet code and does the wiring from it meet NEC standards
There's a concept called grandfathering which says If the work was legal on the day it was installed, then when Code changes later, it's still legal. You're not required to  tear out a bunch of wiring every 3 years when Code is revised.
This gets into a lot of lawyering about when the work was done and what was legal then.  However we can safely pass judgment on things that were never legal.
There are at least 3 panels in this building.   If you look on the &quot;main lugs&quot; above the breakers, you see 2 wires tapped onto those lugs.

One set comes from another location containing a main breaker or main panel, which presumably limits current on these wires to 100A.
Then you have this panel, of course.
Then the other set of wires goes onward to yet another panel.

Those wires are not service wires.  A service is what comes in straight off the meter.  Once it's past the main breaker, it is feeder.   And both panels - this and the next - share that 100A of feeder.  Daisy chaining panels like this is legal, except for...

The most obvious code violation is the double-tapping of those main lugs.  Can't do it. Those 2 wires need to go into a 3-void Polaris connector, then a same-size pigtail needs to come down to the lugs so there is only 1 wire per lug.
Your subpanel doesn't have any ground bar, but that's not unusual when you have a panel that is connected entirely with metal conduit or armored cable, since that metal shell covers the need for grounds.  I do see one ground wire that some epsilon-minus put on the neutral bar; that's not OK and must be dealt with.

To your questions:
#1. Does it meet current Code standards?  It doesn't have to.  It's grandfathered.
#2. Can I simply change a nonconforming panel? You wouldn't need to change the panel; unless you smuggled it in from a third world country, surely the panel is legal stock.  You might have to correct defects, like the double-tapped lugs.
#3. Does NEC limit number of rooms per breaker?  Not as you'd think.  NEC 2020 has requirements for a number of dedicated circuits.  2 dedicated kitchen receptacle circuits.  Dishwasher. Disposal. Built-in microwave.  1 dedicated bathroom recep circuit. 1 dedicated laundry room circuit.   Almost all of them are GFCI or AFCI, and the AFCI must be done at the breaker, unless the route to the first receptacle is in metal conduit.
I don't understand why you're saying that the work has to come up to modern Code.  It doesn't.  The only exception is if the work was done illegally (i.e. without a permit), at which point the city is at liberty to make you tear out all the illegal work (even if there is nothing wrong with it) and then make you pull a permit to re-do the work properly.  Since the permit would be this year, this year's Code would apply.
Lastly, as you have been cautioned, a landlord cannot DIY electrical on a rental unit.  All work must be done by a licensed electrician. It really doesn't matter what your financial situation is; you just can't do it.   That law is to prevent landlords from doing inferior work that puts others' lives in danger.  The logic is an owner-occupant has incentive to do the job properly: a landlord only has incentive to do the work as cheaply as possible.