This varies heavily by country but I can answer this for the United States operating under the FAA/General US legislation (which varies by city).
Here in the US some airports are owned by the local government (there are also privately owned/publicly used and fully privately owned airports). Publicly owned airports (owned by cities/towns/localities etc.) generally either publish their budgets or you can obtain them if you request it. For example if you want to see the information for KPNE or KPHL (owned by the city of Philadelphia) you can submit a form to the city and request the information. Every city is a bit different and there may be small processing fees/long wait times but generally the records can be obtained.
The FAA also published a lengthy brief on the cost-benifit analysis of having an airport in your area. It sheds light on some common costs and such.
Keep in mind that the airport while owned by the city and operated by the city, often has many private companies on its premises that simply lease a building or plot of land. The budgets to operate these FBO's, service centers and the such are more than likely private information.