I think Lillith has hit on a major point. The problem isn't simply the adverb, it's that the adverb doesn't really convey a very specific concept. That can be a problem with a lot of adverbs. As a reader, I'm just not quite sure what is meant by "sat up alertly". But that may also be an issue of context, and your example doesn't include what went before that statement. Is she startled at the time? Awakened? Has she been waiting, expecting something? If you're startled, or awakened, you might be confused, disoriented, fearful, lots of other things. If you're waiting for something, and it happens, you'll have been alert before you sit up. Context matters a lot in finding the precise way to express something like this. And maybe the adverb isn't the best way of expressing that. And maybe it is.
caw