HAMSTER INSPECTOR provided some very good advice in this thread!
Glad your panel was modernized. Sounds like the electrician knew his stuff, I'm sure he took the time to make sure everything was balanced and secured. This is why people preach to always hire a pro for electrical work. I don't believe safety (i.e. fire) is going to be a concern going forward, so please don't worry.
Squash, I'm curious as to the power consumption of your air-conditioner. Is there a metal tag somewhere with it's ratings? Specifically I'd like to know what it's wattage, amperage and voltage (I'm assuming it's 115V, but just in case).
The breaker in your building's basement is designed to protect the conductors between itself and your panel. The fuses (now breakers) in your apartment are designed to protect the individual circuits within your apartment. If a fuse blew, it means that there was excess draw on that circuit (>1800 watts). Since your computer and air-conditioner both died (you mentioned half the apartment), you may have overloaded that circuit - don't worry, the fuse blowing is exactly what was suppose to happen.
This is why I'd like to know your AC's rating (some can be as high as 1500W), and can surge higher at start-up for a few seconds (that's why WhOiSyOuRdAdDy suggested the TD fuses). The computer can be 100-500W, add in a couple of lights and you are close to 1800W. If it happens again, before resetting the breaker, unplug your computer and try to find a plug that still works and leave your computer plugged into it going forward.
Since I believe the circuit was overloaded, this is why your new fuse immediately blew. There were sparks, (most likely) because the fuse wasn't fully seated as you screwed it in and the overload condition still existed. Fuses generally don't smell when blown, so I'm guessing that the wire was overheating and the insulation was beginning to melt. The smell is very strong and can linger for a while.
Please provide the AC ratings and we can confirm or deny my theory.
Not funny.