Use of Lead Screw

[metaltech]

A screw as you describe can lift a lot of weight, or more accurately, generate a lot of thrust. So, you have to have strong mounts to react that thrust. IOW, be able to hold the screw without the mounts deforming. In real machinery, there are bearings designed for mounting leadscrews or ballscrews.

Using something like that would be great, but you may be able to get by with less (cheaper) considering the 'bot doesn't need to run for years. Perhaps a block of nylon with a counterbore (pocket) that the end of the screw sits in. This is NOT optimal (won't guaranty that it will work) but it's pretty cheap and easy to make.