Time to break out some of my sport science literature[1] and what sport science knowledge I've learned so far and try to apply it practically.

Training specifically for strength alone, which is different to power, is at the opposite end of the spectrum to training for endurance. The most important factor in strength training is the intensity of the muscular contractions and not the number of sets and repetitions. Intensity is the amount of weight lifted during the strength training and is affected by a number of variables, but for the sakes of simplicity lets keep it to resistance (weight) loading for now.

In advanced to elite athletes, assuming a constant load\repetition set structure, studies have shown that athletes achieve maximal strength gains by lifting around 80-85% and beyond of their one repetition max (1RM) for up to 6-15 repetitions per set. For anyone unclear what 1RM means, it is the maximum weight you can lift in one repetition of the exercise being performed (e.g. one squat). My prac tutor capped the repetitions per set at six, but the theory-work says that the maximum number of repetitions that can be performed varies due to several factors; such as fibre type composition, the exercise performed and muscle mass involvement.

It is important when training for strength to remember that it is universally recommended to have a rest period of at least, as a minimum, three minutes to allow the ATP and CP energy stores within the worked muscles to replenish, as maximal strength or power development cannot occur with fatigued muscles due to inhibited recruitment of muscle fibres. This is contrary to how one would typically train to improve lactic acid tolerance, where the rest period between sets or intervals can be as short as 30 seconds and gives the muscles little time to replenish their energy stores and remove lactic acid. (This leads to a build up of lactic acid obviously)

I have got my own opinions on what might work best if Cactus-Jack is above being a beginner or intermediate trained individual, as the advice spoken of so far reads as being rather cycling specific Strength-Endurance training, but strength gains are achievable outside of targeted strength training if you are not an advanced to elite trained athlete. Which leads in to my next point, that applying the theory to the bike outside on the road is liable to be less optimal than using a turbo trainer due to the inability to precisely set the resistance, unless you know a road with the perfect steady gradient of just the right length of course.

Since Cactus-Jack is a runner and there is a buzz around tailoring workouts to mimic sport specific movements I would personally say that riding out of the saddle is closer biomechanically to running than riding seated and therefore more beneficial to running, but someone more qualified may disagree with that.

Hard to give a detailed answer without more detailed athlete information, although if you want a generalist rule there is an inverse relationship between the amount of weight lifted and the number of reps performed.

In my opinion theoretically aim for short, sharp, very intense bursts of effort at a high resistance akin to a sprint followed by recovery during the reset phase down to the bottom of the climb in between; should fatigued set in and you are just interested in strength and not strength-endurance benefits do the full 3+ minutes of rest required. How many repetitions to complete? Well without knowing the value of the 1RM just go 100% all out and make it a one rep set...

Due to the high intensity keep the set volume low, probably start at around five or six sets at a maximum to be on the cautious side, per workout as there is also an inverse relationship between intensity and training volume such that high intensity phases coincide with lower volume periods.

[1]Ackland, T. Elliott, B. and Bloomfield, J. (2009). Applied anatomy and biomechanics in sport. Champaign, IL: Human Kinetics.