fix: resolve all failing tests and type errors

[stooit]

Summary

Fixes all failing tests and type errors across the api and shared packages. After these changes bun test reports 22 pass / 0 fail and bunx tsc --noEmit is clean.

Bugs fixed

• Auth middleware case-sensitivity (packages/api/src/middleware/auth.ts) — the public-methods list used lowercase 'post', but Hono's c.req.method returns the uppercase RFC 7231 string. POST /users was therefore never treated as public and wrongly required a Bearer token. Changed 'post' to 'POST'.
• Missing import (packages/api/src/routes/users.ts) — the invalid-body path called badRequest() which was never imported, a guaranteed runtime ReferenceError. Added it to the existing ../lib/errors import.
• Inconsistent shared field name (packages/shared/src/types.ts) — renamed userName to username to match how the API route references the field. No dangling references remain.
• Unimplemented pagination utility (packages/shared/src/utils/pagination.ts) — replaced the throwing stub with a real implementation (page slice, total/totalPages via Math.ceil, correct empty/out-of-range handling) matching the PaginatedResponse<T> contract exercised by the tests.
• Type resolution (tsconfig.json) — added "types": ["bun-types"] so tsc resolves the bun:test globals used in test files.

Verification

• bun test -> 22 pass, 0 fail
• bunx tsc --noEmit -> clean

Constraints honoured

• No test files modified
• No new dependencies added
• Changes minimal and targeted

Assumptions

• bun-types was added to tsconfig.json types as the intended fix for the bun:test global type errors, since it is already a declared devDependency.