We investigated magnetic and thermodynamic properties of S=12 quasi-one-dimensional antiferromagnet KCuMoO4(OH) through single-crystalline magnetization and heat capacity measurements. At zero field, it behaves as a uniform S=12 Heisenberg antiferromagnet with J=238K, and exhibits a canted antiferromagnetism below TN=1.52K. In addition, a magnetic field H induces the anisotropy in magnetization and opens a gap in the spin-excitation spectrum. These properties are understood in terms of an effective staggered field induced by staggered g tensors and Dzyaloshinsky-Moriya (DM) interactions. Temperature dependencies of the heat capacity and their field variations are consistent with those expected for quantum sine-Gordon model, indicating that spin excitations consist of soliton, antisoliton, and breather modes. From field dependencies of the soliton mass, the staggered field normalized by the uniform field cs is estimated as 0.041, 0.174, and 0.030, for Ha, b, and c, respectively. Such a large variation of cs is understood as the combination of staggered g tensors and DM interactions which induce the staggered field in the opposite direction for Ha and c but almost the same direction for Hb at each Cu site.