Is it permissible for a Muslim to supplicate with "O Allah, grant me nearness to You and a good return," or is that specific to one of the prophets?

It is not forbidden to supplicate with this supplication, for it is not something exclusive to Solomon. Moreover, it has been reported that the Prophet (peace be upon him) used to supplicate with it, in a narration transmitted by Al-Daylami from Abu Hurairah.