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

[stooit]

Summary

Fixes all failing tests (now 22 pass / 0 fail) and eliminates all tsc --noEmit type errors across the api and shared packages. Bugs spanned both packages as expected.

Changes

• packages/shared/src/utils/pagination.ts — Implemented the paginate() stub. Handles last partial page, out-of-range pages, empty arrays, and guards against division-by-zero for a zero size.
• packages/api/src/middleware/auth.ts — Fixed a case-sensitivity bug: the public-methods allow-list contained lowercase "post", which never matched Hono's uppercase c.req.method, so POST requests wrongly required a token. Changed to "POST".
• packages/shared/src/types.ts — Renamed the User field userName → username to be consistent with the route handlers and tests.
• packages/api/src/routes/users.ts — Fixed a missing badRequest import and the username field reference.
• tsconfig.json — Added "types": ["bun-types"] to compilerOptions so bun:test imports and process resolve. bun-types was already an installed devDependency — no new dependencies added.

Verification

• bun test → 22 pass, 0 fail
• npx tsc --noEmit → exit 0, zero errors

Constraints honoured

• No test files modified
• No new dependencies added
• Changes scoped to only what the tests require

Assumptions

• bun-types was already present as a devDependency; the type-resolution failures were a tsconfig wiring issue, not a missing package, so the fix was a config directive rather than an install.
• The "correct" field name is username (matching the tests and route handlers), so the shared type was aligned to that rather than changing consumers.