If by tour you mean as a visitor on an underway - you will be given that chance. You will not be assigned as as part of ship's force - you can't qualify as EOOW, get your dolphins, etc.
NR engineers are basically technical follows for the research, development, construction and maintenance of nuclear powered vessels. Other areas they oversee are training, environmental affairs, waste disposal. By technical follow, I mean that your work is basically a watchdog for the two research labs (Bettis and Knolls), procurement agent (BPMI) and various shipyards and vendors that support the NR program. You 'follow' their work and ensure they meet the programatic goals (cost, time, safety, etc.) but don't do any real design work yourself.
As to what you'll be doing, after commissioning, you'll head to Bettis Atomic Power Lab in Pittsburgh for the 6 month Bettis Reactor Engineering School (BRES). BRES is a crash course in the fundamentals of naval nuclear reactor design and construction, including heavy doses of reactor theory, thermo, etc. It is taught at the Master's level by Bettis engineers and scientists. After that you will probably be assigned to headquarters in DC for your first technical assignment. You could also be assigned to Bettis or Knolls as part of the NR contingent at these places. After you have served your commitment, if you're liked, you'll get the option to stay with NR as a GS.
Let me know if you have nay specific questions.