If we have committed a mortal sin, do we have an obligation to schedule an appointment for confession literally ASAP, or does going that coming weekend suffice?
I’ve never called for an appointment, so no experience there. If you check around, though, you may find some parishes (I know mine does) that offer confession before weekday Masses. There have been mornings, when I’ve been there at 6 am - waiting.
I think if possible you shouldn’t leave it more than a few days. So if it’s early in the week you might want to make an appointment.
There isn’t, of course, any set time. I think Our Lord’s words about leaving our sacrifice at the altar if necessary to go and make peace with our brother indicate that its better not to wait.
You should go as soon as it is possible for you to. If it is your normal routine to call for an appointment, you should do so. Otherwise go to the first available public scheduled time, even it it is not your normal parish.
Go as soon as you can by appt or by scheduled time in another parish.
I would go as soon as possible. Remember, we know not the hour when we will be called. I would rather look for a priest and take care of this matter quickly or risk dying with a mortal sin hanging over me.