When leaving the airport building, you can take the airport bus, leaving half hourly from the booth you will see to your right. It will bring you to Baramke, which used to be the long distance bus station, but actually the airport bus is the only one still serving Baramke, now a derelict and empty large square...
You will be 'attacked' by taxi drivers, charging you exaggerated prices for a ride to a hotel. But Baramke is within 5 or 10 mins walk from many hotels such as Sultan, Afamia, Amir, Oriental, Semiramis. I could give you the directions how to get there if you would use the bus. Bus fare is still SYP. 50,- ( unchanged since years ), that is approx EUR 0,70 / USD 1,-.
SYP 600 is indeed the fare for a taxi ride. Within the city NEVER pay more than SYP 50, most trips don't cost more than SYP. 25,-. Insist on using the metre, which honest drivers mostly do without being asked.