fix: resolve failing tests across api and shared packages

[stooit]

Summary

• Auth middleware: Fixed case-sensitivity bug in HTTP method comparison — publicMethods used lowercase "post" but Hono passes uppercase "POST"
• Shared types: Renamed userName field to username in the User type to match usage across all tests and route handlers
• Users route: Added missing badRequest helper function (returns 400 JSON response) that was referenced but undefined
• Pagination utility: Implemented the paginate function to slice arrays and return { data, page, pageSize, total, totalPages } as expected by the test suite

Verification

All 22 tests pass (bun test). TypeScript compilation (npx tsc --noEmit) is clean for source files — remaining errors are only in test files due to missing @types/bun declarations (expected in this setup; tests run correctly via Bun runtime).

Assumptions

• The bun:test and process type errors in test files are acceptable since they only affect IDE/tsc checks, not runtime behavior
• The pagination utility returns 1-indexed pages based on test expectations
• The badRequest helper was inlined in the users route rather than imported from an external module