Transit visa for Serbia

For the issuance of a transit visa, you should have an entry visa for the country you are entering after the Republic of Serbia. In case a visa is not required for that country, you will be asked to present other documents explaining the purpose of your visit there.

http://www.mfa.gov.rs/en/consular-affairs/entry-serbia/visa-requirements

At its meeting of 30 October 2014, the Government of the Republic of Serbia adopted the Decision on visa free entry to the Republic of Serbia for holders of foreign passports having a valid Schengen, UK and other Member States' visa, or visa of the United States of America, and for holders of foreign passports having residence permit in the countries of the Schengen area, EU or the United States of America, which was published in the "Official Gazette of RS", No. 119 of 31 October 2014 and which will enter into force on 8 November 2014.

By this decision, the above-mentioned categories of foreign nationals may, without prior visa application, enter, transit or stay in the Republic of Serbia up to 90 days during a six-month period, but not exceeding the expiring date of the said visas or residence permits.

There is a possibility for foreign nationals to enter the Republic of Serbia with two passports e.a. expired national passport with valid visa or residence permit and valid national passport without valid visa or residence permit.

http://www.mfa.gov.rs/en/consular-affairs/entry-serbia/visa-regime

As stated in Timatic, the database used by Airlines:

Visa required, except for Passengers with a multiple entry or an unused one-entry visa issued by Switzerland, USA or an EEA Member State or:. The visa must be valid for the period of intended stay.

So if you have one of the mentioned visas, you can Exit the Airport. Otherwise, you Need a Serbian Transit visa for that.