The aim of the present work is twofold: On one hand it considers the treatment of time observables in quantum mechanics. In particular, it is concerned with the application of a particular spin-boson detector model to arrival and passage times. Thus, it helps to illuminate the fundamental issue of how to treat time in quantum mechanics. In this course some interesting questions of quantum measurement theory are discussed by particular examples. On the other hand it explores the possibility of extending the quantum jump approach to a model beyond its original, quantum optical framework. It also examines by means of numerical examples whether the quantum jump approach, which employs a continuum of bath modes, promises to provide a good approximation to situations where one actually has to deal with a number of discrete bath modes.