[bubblyworld]

I suspect yes - no need to introduce transfinite ordinals, you simply map from the set Z*, which is the integers but including the nonstandard ones. In fact you don't even need to remove the finiteness hypothesis, the transfer principle should guarantee that every hyperreal has such a representation since you can prove that every real does for the standard version.

(if the finiteness thing seems confusing, remember that there are infinitely large nonstandard integers in the hyperreals, and you can't tell them apart from the others "from the inside")