Learn German: Ich habe keinen einzigen Cent dabei. - I do not have a single cent on me.

While bei mir is grammatically correct and sometimes used, German usually prefers the separable verb dabeihaben (split here into habe ... dabei) to express having something on your person or with you. Using dabei sounds much more natural to native speakers than translating 'on me' or 'with me' literally.

Could I say nicht einen einzigen instead of keinen einzigen?

You could, and people would understand you, but keinen einzigen is the standard, idiomatic way to say 'not a single' in German. Whenever you negate a noun that would normally use an indefinite article, German prefers the negative article kein- over using nicht ein-.

Why does the adjective einzig get the -en ending here?

Cent is a masculine noun acting as the direct object of the verb haben, so it must be in the accusative case. The word kein takes the masculine accusative ending to become keinen. After keinen, adjectives follow the mixed declension rules, which require the -en ending for the accusative masculine form, resulting in einzigen.

Since this is singular, how would the word Cent change if I wanted to say 'fifty cents'?