We introduce fast lattice Monte Carlo (FLMC) simulations, where multiple occupancy of lattice sites is allowed with a proper Boltzmann weight and thus the evaluation of nearest-neighbor interactions can also be avoided. FLMC simulations (with multiple occupancy of lattice sites and Kronecker δ-function interactions) are much more efficient, in the study of equilibrium properties of soft matter such as polymers, than both conventional lattice MC simulations (with self- and mutual-avoiding walks and nearest-neighbor interactions) and fast off-lattice MC simulations (with pair-potential calculations). When compared with the corresponding lattice field theories based on the same Hamiltonian, FLMC simulations further provide a powerful means for unambiguously and quantitatively revealing the effects of long-wavelength correlations/fluctuations. Using confined homopolymers as an example, we report for the first time FLMC simulation data ranging from the single-chain case all the way to the lattice self-consistent field limit.