Celebrim
Legend
This situation isn't imagined in the downtime rules, but if you want to run a contest over a week, simply roll 7x for both the PC and the NPC. Whoever rolls better on the provided tables wins.
Ok, now we are getting somewhere. So, it seems an NPC could roll on the table.
If you want other factors to influence the outcome (skill, some material or social advantage, magical intervention, etc.), then you are totally outside the rules and will need to make a ruling as the DM.
And here is where I find fault with the rules as is. Because even if we don't care what happens with NPCs, you would think that skill, material or social advantage, and so forth should be able to influence the results for the PCs. The guy running a business with the right tool or skill proficiencies or other advantages ought to have advantage on a 'running a business roll'. Using the rules even as is, there is a disconnect from the scenario and the rules we apply to it. Remember when I noted that at a minimum we expect the combat system to advantage people skilled in combat. Well, a downtime system has the same need. Different sorts of choices should be more favorable for different sorts of characters and backgrounds.
But, ok, we could make some rulings that is to say we could make up a rule on the fly, since that is exactly what a ruling is. But the fact that we can make up rulings isn't a feature. The rules don't naturally extend to cover related situations. So they aren't even very good at handling a quick update of the situation of a PC between adventures. They are simply too crude to handle even the basic expected outcome well. They fail for the same reason that Celebrim's 'Simplest Gaming System ever' fails* - the resolution system has no verisimilitude except by accident. Merely being simple and easy isn't enough to make a rule good.
*The core rule (and virtually the only rule) of this system is, "Whenever the player proposes an action, flip a coin. On heads, they succeed. On tails, they fail."