This paper is concerned with monotone (time-explicit) finite difference scheme associated with first order Hamilton-Jacobi equations posed on a junction. It extends the scheme introduced by Costeseque, Lebacque and Monneau (2015) to general junction conditions. On the one hand, we prove the convergence of the numerical solution towards the viscosity solution of the Hamilton-Jacobi equation as the mesh size tends to zero for general junction conditions. On the other hand, we derive some optimal error estimates of in $Lloc\infty$ for junction conditions of optimal-control type.