There is a tendency in Danish for single-syllable nouns to be neuter, although there are no strict rules.
Yes, uge is two syllables (pronounced [ˈuːə]). But again, it could just as well have been a single-syllable word and still be common gender.
As a sidenote, uge is often pronounced with only one syllable as [ˈuː]
Is the "r" silent in the word for "year". If it is, is this unusual, or are final "r"s often silent? Are there regional differences, such as the way the r is pronounced at the end of syllables and words more often in American/Irish/West Country English than in Received Pronunciation English?