Here I was trying to be non-cynical for once.
Of course you can say it after shkia. Erev rosh chodesh sivan may be mesugal for davening for torah, but don’t you think G-d wants to hear your tefillos now also.
Really, it bears noting that this tefilla is not some “segula” (see my published rants). If you look at it, the text is very emotional and sincere. Don’t just zug it; daven it.