fix: repair failing tests and type errors across api and shared packages

[stooit]

Summary

Makes bun test fully green (22/22) and npx tsc --noEmit clean. Bugs spanned both the api and shared packages.

Fixes

• Auth middleware (packages/api/src/middleware/auth.ts): the public-route check compared the request method against lowercase 'post', but Hono surfaces HTTP methods uppercase — so POST /users was never recognised as public. Changed to 'POST'.
• Users route (packages/api/src/routes/users.ts): badRequest was used but not imported. Added it to the existing import from ../lib/errors.
• Shared User type (packages/shared/src/types.ts): field renamed userName → username to match the API contract and test expectations (cross-package field-name inconsistency).
• Pagination util (packages/shared/src/utils/pagination.ts): implemented the paginate() stub (page slice, total, totalPages, pageSize) per the test contract; handles empty arrays and out-of-range pages.
• tsconfig (tsconfig.json): added bun-types to compilerOptions.types (already installed) so bun:test globals and process resolve under tsc. No new dependencies added.

Verification

• bun test → 22 pass, 0 fail
• npx tsc --noEmit → clean (exit 0)

Assumptions

• The canonical field name is username (lowercase), since the tests use it and test files must not be edited.
• Type-only environment errors (process, bun:test) were resolved via tsconfig rather than touching tests or adding deps, since bun-types was already present in node_modules.

Constraints honoured

No test files modified, no dependencies added.