The female name is always Aruna (or Arunaa). For the male, read on.
Depends on the language. `Indian' stands for more than a dozen languages each with its orthography and transliteration scheme, not to mention many who go through their life perfectly happily without ever spelling it in the roman script. In most scripts derived from the Brahmi script (Almost all the `alphabetic' languages of East/South/Southeast Asia except Korean Hangul) the name will have three grapheme clusters: the vowel closed a, the consonant r modified by the short u, and the consonant retroflex n with the inherent closed a sound. The feminine name will put a modifier for open a on the last consonant.
In pronounciation, languages like Bengali and Hindi will drop the inherent a in the last consonant of the last grapheme of the male name, and will transliterate it as Arun. In Oriya, many South Indian languages and Sinhalese, possibly, the closed a in the male name will be pronounced, possibly as the vowel in English cut. The open a in the female name will, by constrast, pronounce it as the vowel in English car (without the r, of course). In this case, depending on the linguistic tradition, both the male and the female name *may* be transliterated as Aruna, but pronounced differently.
The different languages differ not only in the pronounciation of this last consonant. Thus, in Bengali, the first vowel is more like o as the vowel in English cold, the retroflex and dental n are not distinguished, and all vowels have the same quantity. Though I have not actually seen Arun spelt Orun, I have seen similar alterations elsewhere picking up regional pronounciations: Anuttam/Onuttam, Arani/Auroni, Rishi/Rushi, and in fact, my name Tanmoy/Tonmoy/Tanmay/Tanmaya (and other variations, see my namesakes page linked from my home page).
An Indian name is not expected to have consistent orthography in the roman script, and very often it does not have one.
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http://tanmoy.tripod.com/