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

[stooit]

Summary

Fixes all failing tests and TypeScript errors in the multi-package API repo. bun test now reports 22 pass / 0 fail and bunx tsc --noEmit is clean.

Bugs fixed

Area	File	Fix
Auth middleware case-sensitivity	packages/api/src/middleware/auth.ts	Public-method list had lowercase "post", so c.req.method (always uppercase POST) never matched — every POST /users was wrongly 401'd. Changed to "POST".
Cross-package field inconsistency	packages/shared/src/types.ts	User type used userName while route handlers and tests use username. Aligned the canonical type to username.
Missing import	packages/api/src/routes/users.ts	badRequest was called but not imported — runtime ReferenceError on missing-field POSTs. Added it to the existing import.
Unimplemented utility	packages/shared/src/utils/pagination.ts	Replaced the throw "not implemented" stub with a real paginate() (page slice, total, totalPages, empty-array and out-of-range handling).
Type resolution	tsconfig.json	Added explicit "types": ["bun-types"] so Bun globals/bun:test resolve. No strictness loosened.

Verification

• bun test → 22 pass, 0 fail
• bunx tsc --noEmit → no errors
• Independently reviewed and approved.

Constraints honoured

• No test files modified.
• No new dependencies added (bun-types was already present).
• Only changes the tests required.