fix: repair failing tests across api and shared packages

[stooit]

Summary

Fixes all 9 failing tests and eliminates all tsc --noEmit errors across the api and shared packages. Baseline was 13 pass / 9 fail; now 22 pass / 0 fail and tsc --noEmit is clean.

Bugs fixed

• Pagination util (packages/shared/src/utils/pagination.ts): was a stub throwing not implemented. Implemented 1-indexed paging — start = (page-1)*size, end-exclusive slice, totalPages = ceil(total/size) (0 for empty input).
• Auth middleware case bug (packages/api/src/middleware/auth.ts): public-methods list held lowercase "post" but Hono reports methods uppercase, so POST /users was never treated as public. Changed to "POST".
• Shared type field inconsistency (packages/shared/src/types.ts): User.userName renamed to username to match what the API route and tests use.
• Missing import (packages/api/src/routes/users.ts): added badRequest to the import from ../lib/errors (already exported there) — was a runtime/type failure in the 400-validation path.
• tsconfig (tsconfig.json): added "types": ["bun-types"] so process and the bun:test module resolve under tsc. bun-types was already installed — no new dependencies.

Verification

• bun test → 22 pass, 0 fail (37 expect() calls, 4 files)
• npx tsc --noEmit → clean (exit 0)
• Independent review: APPROVE, no blockers.

Assumptions

• Test files are the contract and were left unmodified; source was made to conform (notably the userName→username direction, driven by the tests).
• No new dependencies added; the tsconfig fix relies on the already-present bun-types.