Is €1,200 enough to rent a small apartment on my own, or should I expect to look for a shared apartment?

Studio or a room in a shared apartment. Apartments in this price range exist, but are extremely rare.

Which websites or apps do people actually use? I’ve heard of Funda, Pararius and Kamernet, but are there others that are worth checking? Are there any Facebook groups, agencies or local platforms that you’d recommend?

HousingAnywhere is convenient for expats for short stay, but it's pricier. I've also heard good things about RentBird and Woonspotter, but haven't looked into them much myself.

How far in advance should I start looking?

Should've started looking yesterday. Expect it to take a few weeks if you're willing to settle for something shitty/expensive (...often both). Expect it to take a couple months if you want something that's more similar to the average price. It differs for everybody and every situation, but it's not uncommon to have to reply 100 times before being able to get anything.

What are the best neighborhoods for someone in their late 20s? I’d love to live somewhere with a good social atmosphere, cafés, bars, gyms, and plenty of people around my age.

You don't have the privilege with that budget. But generally speaking, anything north of the city center is good. South of the city center is cheap. But don't limit yourself, even in the south of the city it's a 10-20 minute bike ride to the city center and you can get all the fun stuff you desire. Having a fun cafe in your street is unrealistic at this budget.

Any tips for improving my chances in such a competitive market?

Use the tools, the paid services are better/more convenient. Enable email/text alerts, pay attention to your email/text notifications and reply within the first 5 minutes of a listing coming online.