fix: make all tests pass and eliminate type errors

[stooit]

Summary

Fixes all failing tests and TypeScript type errors across the api (Hono) and shared packages. Final state: bun test → 22 pass / 0 fail, npx tsc --noEmit → clean (exit 0).

Bugs fixed

1. Pagination not implemented (packages/shared/src/utils/pagination.ts) — replaced the throw new Error("not implemented") stub with a real paginate<T>(): 1-indexed page, totalPages = ceil(total/size) (0 when empty), out-of-range pages return empty data, correct partial-last-page slicing.
2. Auth case-sensitivity (packages/api/src/middleware/auth.ts) — the public-method check compared a lowercase literal against Hono's uppercase c.req.method, so POST /users was wrongly treated as protected. Normalised the comparison to be case-insensitive.
3. Inconsistent shared field name (packages/shared/src/types.ts) — User.userName renamed to username to match the tests and usage across both packages.
4. Missing import (packages/api/src/routes/users.ts) — badRequest is now imported from the existing ../lib/errors helper (no new dependency).
5. Type errors for process / bun:test (tsconfig.json) — set compilerOptions.types to ["bun-types"] so the already-present ambient Bun/Node types resolve. No dependencies added.

Constraints honoured

• No test files modified.
• No dependencies added (package.json untouched).
• Minimal, scoped changes (5 files, +10/-9 lines).

Assumptions

• Pagination is 1-indexed and returns empty data for out-of-range pages (from the pagination test contract).
• username (lowercase) is canonical, since test files (uneditable) use it.
• page<=0 / size=0 are not guarded — not exercised by tests, not wired to a live route.

Verification

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