Cinquante-et-unième because État is masculine and therefore un has to be too.

Regardless of the gender of the word it applies to, 51^e is written cinquante-et-unième and that last component is never pronounced “un – ième” in French, i.e. *[œ̃ jɛm] or *[ɛ̃ jɛm].

There is no nasalisation so it is pronounced like “une’ième“: [ynjɛm]