How to fix electrical system that feeds a studio unit
First, that 20A dual breaker does not have common trip.  It needs common trip.  Get a 2-pole breaker (about $10) or a handle-tie (about $3). 

That thing you have pictured, the tiny Cutler Hammer (BR?) panel, is your main breaker box, since it's the first stop from the meter.   Here, and only here, neutral should be bonded to ground.  

This is actually rather good news.  It is not realistic to DIY replace a main panel, because there's no way to completely de-energize it. By having this "runt" main panel, you will be doing pretty much all the house's wiring in a sub-panel which will replace the fuse box.  And that makes DIY easy and safe, just shut off the main breaker in the exterior panel. 

Get yourself a nice 40/42-space subpanel and you're good to go.  (you can get away with a smaller subpanel, but that is no place to scrimp - ask anyone who has run out of space in their panel.) I suppose you could do a 30, but you'll have a happier kitchen if you give it decent electrical service - one circuit per half-receptacle is not excessive, ask any cook who's tripped a breaker!  These bigger panels tend to come with a lot of free "bonus" breakers, and that helps with the finances. But the main criterion is sanely priced AFCI and GFCI breakers - because many of these will be required in the future.  Since AFCI and GFCI are full-width, "half-width" or "duplex/twin" breaker schemes are useless, so avoid "20 space/40 circuit" panels.  

You may think it odd to hook a 225A subpanel to a 20A breaker.  That is actually fine because the 225A subpanel can handle 20A.   (a 60A subpanel off a 100A breaker, no.)  Given that all modern service must be at least 100A, the city will probably require 100A at the very least, the reason to go 150++ is more spaces, and more future expansion. 

Why would you ever want more amps?  Four words, On-demand hot water which are great for studios because they're small. Some fine ones can take 120 amps all by themselves (but 0A the rest of the time, hence the cost savings.)  Also the usual - electric range, dryer, EV charger, etc. There's always another thing. 

In the future, you upgrade by contacting the power company to talk about your options (your subpanel size is a limiting factor here, if it's 100A you cannot get 200A service etc.).  Then you obtain a new meter pan with main breaker onboard, and run appropriate sized wiring to your fine subpanel.  Or you could run that wiring to your current mini-main panel today, it's aluminum wire and not terribly expensive.  (this aluminum wire is fine.)  You would need lug splices to go from the fat stuff to a smaller wire size that would fit in a 20A breaker, but that's no big deal.