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

[stooit]

Summary

Fixes all 9 failing tests and all tsc --noEmit type errors in the multi-package (Hono api + shared) repo. After these changes bun test reports 22 pass / 0 fail and npx tsc --noEmit exits clean.

Changes

• Pagination (packages/shared/src/utils/pagination.ts) — implemented the previously-stubbed paginate() (it threw not implemented). Correctly computes total, totalPages, page/pageSize, and the page slice; handles empty arrays, out-of-range pages, and partial last pages.
• Auth middleware (packages/api/src/middleware/auth.ts) — fixed an HTTP-method case-sensitivity bug so the public-route check matches regardless of method casing, making POST /users public (no token required).
• Users route (packages/api/src/routes/users.ts) — added the missing badRequest import that was causing a runtime failure (and TS2552) on the 400-for-missing-fields path.
• Shared type (packages/shared/src/types.ts) — aligned the User field name to username for cross-package consistency (resolves TS2561); the field name is now canonical across both packages.
• tsconfig — added a types entry so bun:test and the process global resolve without installing new dependencies.

Verification

• bun test → 22 pass, 0 fail
• npx tsc --noEmit → clean (exit 0)
• Independent review pass confirmed pagination edge-case math, the auth casing fix, and cross-package field-name consistency.

Assumptions

• No test files were modified and no dependencies were added, per task constraints.
• Existing type packages in node_modules were wired up via tsconfig rather than installing anything new.

Constraints honoured

• No test files modified · no new dependencies · minimal, targeted fixes.