Solved – Cubic and linear relationships in multiple regression model

One way to fit the model is, as you guess in the comment, to transform $X_1$ first, then run a multiple linear regression. However, you don't have to do any transformation back to the predicted $Y$ value, since the regression is still using the untransformed $Y$ variable as the dependent variable. Let's say you create $Z = X_1^3$, then the regression becomes $Y = beta_0 + beta_1 Z + beta_2 X_2 + e$. The estimates and/or predictions you get out of the model with the transformed $X_1$ are still for $Y$.

There are other ways to fit the model, but the choice between fitting methodologies is, in this case, independent of the transform of $X_1$.

Edit: It has occurred to me that you may have meant a cubic relationship where $Y^3 propto X_1$. This is equivalent to $Y propto X_1^{1/3}$, and you can still just transform $X_1$ without having to transform $Y$ at all.