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

[stooit]

Summary

Fixes all failing tests and type errors in the multi-package repo. bun test is now 22/22 green and bunx tsc --noEmit reports zero errors.

Changes

• packages/shared/src/utils/pagination.ts — implemented paginate() (was a throw new Error("not implemented") stub). Handles empty arrays, out-of-range pages, and partial last pages; returns data, page, pageSize, total, totalPages.
• packages/api/src/middleware/auth.ts — fixed a case-sensitivity bug where the public-methods list contained "post" (lowercase). Hono's c.req.method is always uppercase, so POST /users was incorrectly treated as requiring a token. Changed to "POST". Also resolved the process type error via tsconfig wiring.
• packages/api/src/routes/users.ts — added the missing badRequest import that caused a runtime failure on the 400-for-missing-fields path; updated field usage to username.
• packages/shared/src/types.ts — renamed User.userName → username to match both the tests and the route handler (the type was the outlier causing the cross-package inconsistency).
• tsconfig.json — wired in the already-installed bun-types so bun:test and Bun globals resolve under tsc (no new dependencies added).

Verification

• bun test → 22 pass, 0 fail
• bunx tsc --noEmit → exit 0, no errors

Assumptions

• Field naming standardised on username because the test files — which must not be modified — use username.
• No test files touched, no dependencies added; changes are minimal and scoped to what the tests require.
• Negative-page input to paginate() is out of scope (not reachable via any route, not exercised by tests).