How much does Enhanced Token Info cost?

The DEX Screener marketplace lists $299, discounted from $499.

Is there a free way to update token info?

Tokens on external lists such as CoinGecko's are picked up automatically, and token pages have an Update Token Info link.

Automatic. DEX Screener looks for token information in external token lists such as CoinGecko's. Once your token is on one, its info appears on DEX Screener automatically. The help article that says this is old, so check it still applies.
Enhanced Token Info (paid). The docs say this gets your info on screen as quickly as possible regardless of listing on third-party services. The marketplace page lists it at $299, discounted from $499. Questions go to support@dexscreener.com.
Update Token Info link. Token pages show an "Update Token Info" link and a prompt to claim the token profile. This is the likely start of a free profile submission, but the steps behind it are not documented in the pages we reviewed.

Moonshot launches list Enhanced Token Info as included at no cost.

Prepare a description, website, social links and a logo. Related: how to get your token on DEX Screener and DEX Screener.