First, I would like to say that the book you got is actually not really a tactics book. It is much more than that. It has chapters on opening play, on positional topics, on endgames and, yes, also many on tactics. I really like the book and I feel that working through it has helped me a lot.
Essentially, Yusupov gives you all the advice you're asking for:
1) play through all the variations on a board and try to understand all the ideas in them.
The book is very well structured so in many cases there are really not too many things going on simultaneously, at least in the earlier chapters. It is certainly a good idea to try and see whether you can even solve the examples that are used to explain the concepts. At least spending a couple of minutes on each before looking at the proposed solution and the variations may help you to pick up the main points more quickly. On the other hand, if you are not at all familiar with the concepts that are taught in a chapter, this may not make too much sense.
2) solve the exercises without moving the pieces and write down all necessary variations, only then check and compare with the solutions. If you cannot make progress without moving pieces, then try with moving pieces for a couple more minutes.
I feel that this writing-down process has really helped me to find more defensive option of my opponent. Also, whenever I got really stuck on one of the exercises, it was because I had either missed an important idea or I had stopped calculation of a variation too early without being able to assess it properly. Both give you a more or less direct push into a direction for improvement.
I am not sure it really has to be a real board. When I am traveling, I often use an ipad for this as it is simply more flexible. I feel that I have not too much trouble moving back and forth between this and a real board. In particular, if you use functionality for saving and replaying variations you need to be careful to really look at each move and understand its purpose. It can be rather tempting to simply play the animation and look at the fireworks on the board without understanding what happens. While this will consume valuable chess training time, it will likely not lead to a substantial chess improvement.
A final, suggestion from myself would be not to rush it. Take your time really working through the book. I actually went over it a couple of times, redoing the exercises. Since there are 12 per chapter I have not fully memorized them, so in many cases I still have to find the solutions. Sometimes it then happens that I remember that the move I just found is actually correct. I then force myself to really check all different variations as I would have to in a real game.