I hardly ever talk on the phone until late at night.
Literal
Night late [until-まで] phone [with-で] talk thing [etc-など] hardly rarely not-exist.
~など is a softening/listing particle — 'things like X.' The frame まず (here in the adverbial sense 'hardly') + めったに~ない ('rarely') stacks two negative-polarity adverbs for strong emphasis. The double-negative intensifier is a common written flourish.