You can, but here's what will happen: Since this is not the computer you normally sync with, all iTunes content will be erased from your phone. You can re-download purchased content, but non-iTunes content you'll lose, as well as any synced photos.
You'd need to first authorize this computer: iTunes>Store>Authorize this Computer. Then, iTunes>Preferences(under the Edit menu in Windows)>Devices...put a check in the box to prevent iPods/iPhones from syncing automatically. Connect your phone...DO NOT SYNC...right-click in the left device pane, select Transfer Purchases; next, right-click again & select backup. Then, follow the directions in that support article & restore your phone from the backup you made when you started, sync or re-download your purchased iTunes content.
When you get back home, you'll have to go through this again...remember, by design, iPhone syncs or manually manages iTunes content with one computer at a time.
Whatever computer you use, make sure you setup a separate user account & use that account to do this. DO NOT USE any existing user account on any computer you chose.