You can check a couple of options:

Look at flying into Istanbul, and then flying on to the Turkish side of Cyprus. The main airport in Turkish Cyprus is Ercan (ECN). Turkish Air, and some low cost Turkish carriers such as Pegasus, Atlas Jet, Onur Air, and I think Sun Express fly the route.

Ferriers into Cyprus come from Alanya and Mersin Turkey, and those options would be no cheaper than flying from Istanbul.

Wizz Air has some fairly inexpensive flights into Larnaca from Budapest. Ryan Air flies into Paphos. And, EasyJet flies into both Paphos and Larnaca.

Unfortunately, you're flying in August, and it's prime tourist season. You'll love Cyprus, it's a great place.