I have a situation where a character is locked in a cell for about two week’s time. This is a medieval setting. For plot reasons, the character is chained to a block in the middle of the cell. The problem is, I have no idea how he would pee and such. I find it really awkward to think about, and I don’t think it’s relevant to the narrative to explain it, but I’m sure the question will come to reader’s minds, and it does slightly bother me that I don’t know the answer. Usually grazing over details is lazy, but I really don’t want to confront this one. Do I have to come up with a solution, or will readers generally not care?
Generally, most readers won't care... and the ones who do are weird. "Going to the Bathroom" is only brought up for low brow humor or plot points (such as a detail of an escape). You can even draw attention to your omission by having a casual question tossed to the character ("Wait... how did you go to the bathroom?" met with "I'd rather not dwell on that.") OR you could have the character ask this upon arrival in the cell, only for the guard to say nothing and give a look that the guard is being silent to be cruel, but also trying to hide his giddy humor in keeping that knowledge to himself.
Either way, the options both address the issue and don't, and will let your readers come up with their own ideas. Sometimes the best horror is not knowing.
This is a slightly obscure question, and might be better answered someplace like the History SE or worldbuilding.
It's more relevant if the character isn't from the period (like a time traveler), but the logistics of it won't be such a big deal for folks of the period. No one has felt the need to answer this question in books I've read unless waste disposal was somehow relevant to the story or plot. If you WANT to be dramatic or add color, the classical thing to say is that the person sat in their own filth. It's worse if the chains don't allow someone to take down their own pants.
In case it IS relevant to the plot, and assuming minimal sanitation, it will be either a bedpan (to keep the cell clean) or it's the floor (if filthy is part of the punishment). Solid waste is problematic, but a dirt floor will soak up urine eventually, and smell bad to boot (manacles were useful because prison cells were poorly constructed and vulnerable to digging and escape attempts). Only the smell would be as much a punishment for the guards as for the prisoner. But remember that this is a time when people emptied their waste into the streets where people walked.
Generally, the hygiene level is not high for the period (varying from region to region), and especially for prisoners. Lots of prisoners died of the horrible conditions in jails, one of the reasons physical punishments like beatings or executions were common (and housing prisoners is expensive).
It depends on whether leaving out details would constitute a plot hole
There is no problem in leaving out details - if those details are not very relevant. Many authors tend to omit "dirty" details related to bodily functions, and if your book, in general, is not very descriptive and realistic, this would be fine to omit those specifics too.
However, sometimes those details can be of higher importance, and at some point, any attentive reader can't help it but ask "how?".
For example, how would Edward Scissorhands go to the bathroom?
If there is an obvious solution to the problem, for example, the one which @DWKraus' answer suggests, it would be totally fine for you to omit details and leave it all to the reader's imagination. But if not, your entire book should not be based on realism, otherwise, you are creating a plot hole.
You have to come up with a solution.
Whether you share that solution with the reader depends upon the genre, audience, and historical accuracy; whether the solution is an opportunity to show the personalty and motives of your characters; and whether who is telling the tale of the imprisonment would share these details.
I do not think much detail is needed, "imprisoned for two weeks" already tells us that this a political prisoner. Water, food, cleaning of defecation, the care of festering injuries, is either all dealt with by the imprisoner (or the family of the prisoner who bribe the imprisoner) or are all ignored by the imprisoner. The prisoner is either kept in luxury til ransomed, or is to be killed by omission through starving and infected injuries.
In classic stories, like the tale of Prometheus (which I assume is the blueprint for this), this question is left out because it is unnecessary - you also wouldn't explain how the character took a breath or grew hair (unless it's a wizard beard)
The classic story also had an awkward part where a bird eats Prometheus's liver each day, and no one cared if this was awkward. The only reason modern humans don't want to think about "awkward" stuff (your wording) like this is that we somehow tell ourselves that we are above these filthy natural things - but this could be a real story boost!
Think about it - is your character a nobleman who never actually had to cook for himself? Maybe this is the chance for character growth in general. Like "I had to sit in my own filth for 2 whole days before somebody came and cleaned after me, I came to realize all of humankind was always sitting in their own filth, no one is better than anybody."
I feel this is obligatory: https://youtu.be/rEMjRmKViZo
Depending on the tone of the story, there are no limits to what you can get away with.