Try the Academie Ecole de Coiffure, Rue de Grand-St-Jean. A men's cut and wash is 20Fr and you can go without appointment. It's an academy, so you will have a trainee cut your hair but I've visited twice and been satisfied with the results both times. As they are students, they take their time, and the teacher will have a look at the end to tidy up if necessary.
If, like me, you don't speak great French, take along a photo on your phone to show them what you want exactly. http://www.academiedecoiffure.ch/