gramatická analýza: (Nejdřív jsem běžel) k té přejeté.

Past Passive Participle long form přejetý from přejet

The main verb (the conjugated verb) is jsem běžel; the subject of this sentence is therefore já. Being in the nominative position, já cannot be modified by přejetý which is governed by the preposition k which always takes the dative case. Looking at the ending in the dative, you can determine that přejeté cannot be other than fem. sg. dat (there is no dat. pl. form of an adjective ending -é). The PPP thus is modifying an implicit feminine noun žena, stará žena or stará žebračka (the woman/old female beggar who was run over (by the car)). Thus, the sentence could be paraphrased as "First of all I ran to that woman whom the car ran over."