In order for this to work, there are a few 'musts' that need to be maintained.
A male must be kept healthy.
A sick or unhealthy male is a male that cannot perform. Therefore, he must be kept in a place that is clean. He must be provided with plenty of healthy food. He will need time in the sun, so his body can synthesize certain vitamins it cannot otherwise produce. He will need exercise other than sex to maintain his body's systems and keep his weight at a health level. Research done in Australia found that overweight men were less fertile, so his weight level would be policed as well.
If a male becomes sick, this would be cause for something of a panic within the culture...particularly if there are no current replacements available. As a result, for humans, Medical science would be rather advanced. In a typical society of mixed genders, if some breeding groups die due to disease, it is an acceptable loss. In this society, losing a male to disease would be a crushing blow and could, in some cases, wipe out an entire society.
Additionally, a male must be kept mentally healthy as well. Humans need social interaction and to be treated well in order to stay mentally healthy. A male's behavior would be closely watched for signs of instability. Social Isolation is dangerous and can lead to all sorts of problems. Mental state, particularly depression, can have a negative impact on fertility.
A male must not be injured.
In this society, males are the most valuable thing in the world. Without them, a culture can only survive a single generation. Harming a male in any way would be an extreme taboo and harshly punished. Even if battles were fought, a male would be protected from harm by both sides at all costs.
The only time this may be breached is in cases of extreme war where you need to cripple your enemy long-term, and cannot possibly abduct their males. Such an act would likely be viewed as extremely barbaric. It's essentially an act of genocide.
A male must be kept alive as long as he remains viable.
This ties in to the two above points, but since males are so rare and valuable, the longer you can keep one alive and viable for reproduction, the better
The male's busyness will depend on the size and tech-level of the culture
In the modern world, the Crude Birth rate per thousand people is 19.4, globally. This means that out of every thousand people, roughly 19 of them have a child each year. To maintain population stability, this number goes up if there is a war, or famine, or if the tech-level is lower than modern times. The current highest is in Niger, at 51.26 births per thousand people.
So, if you assume an average level birth requirements, if a male fathers 20 babies per year per thousand people in the culture, the population will stay stable. So, how he is treated will depend on how big the culture is...if its only a town or tribe, the male may have plenty of freedom and a pretty open schedule to spend how he wishes. If you rank things up to 'nation' level, things get a bit hairier for the guy.
Taking the completely random example of Ireland. Ireland has a current population of 4.595 million. In order for the population of Ireland to maintain the average global birth rate, our poor male would have to father 91,900 children per year. Or, 251.78 per day. This is a statistical impossibility, so we can see here that society size would be limited by the number of children that the male could feasibly father.
Scheduling for optimal society size
As mentioned above, there is an upper limit to how large a society could grow from the efforts of a single man. So, how would the life of a single male be scheduled for optimal breeding efficiency, given the requirements listed above. To maintain optimal health, 8 hours of sleep per night is required. It is recommended to get a minimum of 30 minutes exercise per-day, and our guy is still going to need this, because sex doesn't exercise all the muscles. He's also going to need meaningful social interaction that is not just the hello/goodbye of a scheduled sex meeting. He's going to need some friends to talk to. At least an hour or two per day. We'll say two hours, to be safe. Additionally, assume he needs a total of an hour to eat three meals in a day. Tack on an extra hour and a half for cleanliness and a bit of 'me time' for him, and that leaves us with 11 hours per day to 'work.'
So, assuming an 'average' young guy with an 'average' ejaculatory latency time of 2-4 minutes, plus some 'hello' time, some time to get started, and some time to clean up...we might be able to pack things in to about 15 minutes for a full turn-around. While there are no concrete studies around odds of pregnancy from a single act, I ran across an uncited statistic that stated that each time egg meets sperm, there is a 25% chance of fertilization. Assuming the female scheduling is managed well (explained below), we can thus assume that 25% of his encounters will result in a pregnancy.
So, 11 hours, 15-minute turn-around, 25% success rate. This means that we are looking at a maximum output of 11 babies per day. Factor in a ~9% chance of miscarriage (also assumes optimal scheduling and planning) or otherwise failed pregnancy, and we are looking at a yearly crude birth rate of 3,653.65. At the modern average, this means that he could support a population of 73,073.
In order to maintain an optimal society size, a woman's encounter with the male would have to be planned and scheduled as well. A woman reaches full fertility at about 19 years old, and maintains a high fertility until their mid-thirties. Likewise, the percent chance of miscarriage also rises after the woman hits her mid-thirties. Additionally, a woman is at peak fertility roughly 14 days prior to her next scheduled period. So, if you are seeking to have an optimally large society, women 19-33 would be ideal, scheduled to meet with the male 14 days prior to their next period. There are other biological indicators that can be measured to detect a woman's actual place in their cycle (look up 'The Rhythm Method' for more details)
How we arrange the meetings
There are really two ways that the appointments can be met by the male. Either the male travels between locations, and you sign up for your appointment with him while he is in town. Or the male stays in one place, and you come to him. For purposes and safety and security, the latter option makes the most sense. This means that women of age would travel to the capital to see 'the male.' However, since women's menstrual periods tend to sync up when they live together, you would need to keep the 'pilgrims' living separately so that their cycles would stay spread out.
As for the details
Given the 'musts' listed above, there is some flex room. Does the culture want to be as large as possible? What measures do they take to keep their male(s) happy and healthy? Do they indoctrinate them and basically brainwash them into being happy with their lot in life? Do they educate them and give them all the comforts they could want? If a male 'doesn't want to' at some point, does he have a choice? These are all flexible things, but the stack of requirements above holds true.