We construct some examples of special Lagrangian submanifolds and Lagrangian self-similar solutions in almost Calabi-Yau cones over toric Sasaki manifolds. For example, for any integer g≧ 1, we can construct a real 6-dimensional Calabi-Yau cone M_g and a 3-dimensional special Lagrangian submanifold F¹_g:L_g¹→ M_g which is diffeomorphic to Σ_g × R and a compact Lagrangian self-shrinker F²_g:L_g²→ M_g which is diffeomorphic to Σ_g × S¹, where Σ_g is a closed surface of genus g.