C. Timely is probably suggesting the inn was reached just in time.
In the phrase "the timely inn", timely is being used as an adjective to describe the inn. This eliminates option A. The speaker is describing the inn as being timely because the traveler needs a place to stay before it gets dark. Just as the light begins to fade, the inn appears, just in time. Options B and D are incorrect because the inn does exist and timely doesn't not describe how something looks.