Deciding whether to charge hourly for a messy maintenance job

A client has a messy maintenance job: update old pages, fix whatever breaks, and investigate a few issues that may or may not be related. I can't reasonably estimate the total hours before opening everything up. A fixed quote protects the client but could punish me for surprises; hourly billing feels vague unless there's a cap. Has anyone used a paid diagnosis phase followed by a firm estimate for the actual repairs? That seems fairer to both sides than guessing.