When does your nose tip start to weaken and droop? Around what age?

This is a very individualized process that depends on a number of factors most of which are genetically determined:  your cartilage strength and thickness of the skin are the two most important.  Likely by early to mid 30s there is some weakening of the supporting ligaments between the cartilage that results in early tip droop.

 The nose tends to age  by thinning of the skin and weakening the ligaments connecting the nasal cartilages. When patient's have an  Active depressor septi ligament, every time they smile it pulls the tip down which further weakens the droopiness over decades.

As we age, the nasal anatomy changes and the tip may start to droop. In addition I have seen patients nasal anatomy change becoming twisted or develop a dorsal hump. These features may occur as early as your 40s or 50s. Over time they become more exaggerated.