If you intend to manually enter every task start and finish date, I would suggest using manually scheduled tasks and set the Project calendar to the 24 hours calendar.

Whether Project is the "right" tool depends upon what you need to do. You can treat Project like Excel using manually scheduled tasks and typing everything in. If you want project to do more it certainly can. It is a scheduling tool - so by creating links (predecessors and successors) along with resources you can create a dynamic schedule that if Task A is linked to Task B, C, and D - the schedule of B,C, and D will change if A is changed.

However, it takes effort to set up the Project file from the start - longer than 30 minutes. The payoff is in update and maintenance of the plan in Project plus the automatic scheduling. I suggest a good hands-on minimum 2-day class.

Let me know the details of your shifts (which days per week, do they take breaks, etc) and I'll walk you through step-by-step.