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Is there an air service to Szekesfehervar from Budpest?
Is there air service to Bratislave from Szekesfehervar?
As stated above, use the train. It's the most convenient way due to short distance.
It's really a short distance. Now there is a track reconstruction going on between Budapest and Szekesfehervar, it causes delays, but you can still make it in one hour. Trains depart from Budapest-Deli or Budapest-Keleti station, both are accessible by metro M2. If you're coming from Budapest-Ferihegy Airport, you may consider taking a coach to Szekesfehervar. The journey time is about the same (it depends on the actual route, though), the ticket costs slightly more than the train. The advantage of the bus is that it departs from Budapest, Nepliget (metro M3 Nepliget station), which is closer to Budapest-Ferihegy Airport. And its stop in Szekesfehervar is in the center, while the train station is a 10-15-minute walk or a 1.5-2€ taxi ride. But you'll have less space, no restroom and your baggage will not be with you but in a separate compartment. Bus and train frequencies are at least one vehicle per hour, but generally it's more frequent than that.
Travelling from Bratislava to Szekesfehervar, you'll have to go via Budapest, due to a lack of direct connections and main north-south routes. Still you may consider taking a train from Bratislava to Komarno, cross by taxi (2-3 euros) or on foot (20-30 minutes) if you don't have much luggage and take a bus from Komarom to Szekesfehervar. I didn't check whether it is shorter in terms of journey time than travelling via Budapest but it's definitely cheaper. But check schedules, these buses do not run frequently (5-6 times a day).
Check schedules here:
International schedule search site of DB German Railways www.bahn.de
MAV-Start (Hungarian passenger rail carrier) www.mav-start.hu
Volan Association (organisation of coach operators, schedule)
Slovakian train and coach schedule search engine www.cp.sk
Budapest Transport Limited (city transport) www.bkv.hu