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Using the first-principles density-functional calculations, we investigate the electronic structure and magnetism of the kagome superconductor CsCr₃Sb₅. At the ambient pressure, its ground state is found to be 4×2 altermagnetic spin-density-wave (SDW) pattern, with an averaged effective moment of ∼1.7μB per chromium atom. The magnetic long range order is coupled to the lattice structure, generating 4a₀ structural modulation. However, multiple competing SDW phases are present and energetically very close, suggesting strong magnetic fluctuation and frustration. The electronic states near the Fermi level are dominated by Cr-3d orbitals, and flat band or van Hove singularities are away from the Fermi level. When external pressure is applied, the structural modulations and the energy differences between competing orders are suppressed by external pressure. The magnetic fluctuation remains present and important at high pressure because the non-magnetic phase is unstable up to 30 GPa. In addition, a bonding state between Cr-3d_xz and Sb^II-p_z quickly acquires dispersion and eventually becomes metallic around 5 GPa, leading to a Lifshitz transition. Our findings strongly support unconventional superconductivity in the CsCr₃Sb₅ compound above 5 GPa, and suggest the crucial role of magnetic fluctuations in the pairing mechanism.