Construct a homeomorphic map

Hint: Think about radial projection from punctured Euclidean space to the sphere. The fibers of the projection are radial rays. Each of these rays is homeomorphic to the real line. Each ray corresponds to a unique point on the sphere, and vice-versa.

Also, don't you mean $S^{n-1}$?

Define continuous function $f:\Bbb R^n\backslash\{x\}\to\Bbb S^{n-1}\times \Bbb R$ by $$f(y)=\bigg(\frac{y-x}{||y-x||},\log_e(||y-x||)\bigg),\forall y\in \Bbb R^n\backslash\{x\}.$$

with continuous inverse $g:\Bbb S^{n-1}\times\Bbb R\to\Bbb R^n\backslash\{x\}$ by $$g(v,r)=e^rv+x,\forall v\in \Bbb S^{n-1},\forall r\in \Bbb R.$$