May be a problem. In most of the cities the downtown is old, generally built in XIX century, parts of it probably older. This means the streets are designed with the needs of those times, not cars. Even around novogradnja, if it was built in the sixties or seventies, there may not be enough parking space.
So you need to perfect your skills with parallel parking, usually with the right wheels on the curb. Smaller cars do it easier, so think of this when renting.
In practically every downtown around here, the streets are zoned, and every few dozen meters, usually near a corner, there's a sign which says which zone it is. Depending on the zone, you may leave your car for 30, 60, 120 or 180 minutes. In some places there are attendands checking your tags, in many there are cameras and the attendant is sitting somewhere and watching the monitors.
To pay, you need a cell phone: on the sign there's a 4-digit phone number (beginning with 8) to which you send a SMS message with your tag number, and you get a message (sometimes) confirming your payment. This is usually about 25-30 RSD. Ten minutes before your hour expires, you get a warning message. Depending on the zone, you may extend it once or twice.
If you park where you shouldn't (or stay longer), it's quite possible that the owner of the parking will call the truck (called "pauk" - spider) which will lift your car and take it away. They don't tow, because the tow truck needs to get in front of your car, which is often unfeasible, so lift it is.