Bewildered word usage

Yes. Bewildered does not mean what you think it does. You might mean confused, unsure, or stumped, or even unable. The phrase "to be able" is also misplaced. The "what his age was" phrase is technically correct but awkward and not really the kind of thing a native speaker would say.

Your sentence would be grammatically correct if it were changed to -

A youthful smile spread on his handsome face, and I was unable to determine his age.

Check Dictionary.com as it is a decent (albeit imperfect) resource, in order to determine what various words mean in English. While bewildered does mean perplexed, it tends to have a connotation (here's where Dictionary.com falls down on the job) of a deeper level of confusion. What I believe you are going for is someone who is slightly unsure but not so confused that they barely know which end is up. To get a better idea of shades of meaning among synonyms, use a thesaurus.

I took a thesaurus bone to show and tell once.

Teacher said it was from a rabbit. I didn't realise that rabbits had been around that long.

Wolf, Blackie is right unless what you really mean is you were surprised to discover than you were able to guess his age