Does an old outlet need grounding?

It's totally worth switching to grounded outlets, especially if you're thinking about using more powerful appliances or sensitive electronics. Two-prong outlets are pretty old and don't have protection against electric shock if something goes wrong. The ground wire is basically a safe path to discharge energy in a controlled way if there's a problem, like a wire melting or someone touching somewhere they shouldn't. With three prongs you can safely connect your fridge, washing machine, AC, gaming console, laptop charger... everything way more safely. There are old outlets that work fine, but it's like driving an old car without airbags - technically it works, but if something goes wrong you're screwed.

The main thing is that for a grounded outlet to work properly you need an actual ground wire running through your wall wiring, it's not just about putting in a new outlet and calling it done. It depends on how your apartment is wired. Some buildings were built before grounding became mandatory, so the wiring doesn't have that third wire. In that case you'd need to call an electrician to check it out and redo the wiring, which costs more but it's an investment that makes a real difference. If your building already has grounded wiring installed, then it's easy - you just swap out the outlets and the electrician can do it in a couple hours.

My suggestion is to call a professional, like an actual electrician, and have them check if there's already grounding there or if you need to install it. It's not expensive to get an assessment and then you can make the decision with proper information. If it's just swapping outlets it'll be cheap, if you need to redo the whole wiring it'll be more of a hassle but it's still worth it for the safety.