In a 2v2 game, can you ready up your partner's runes with Battle Preparations?

Yes — you can ready your partner's runes with Battle Preparations.

The card just says "ready up to 2 runes," with no controller restriction, so any runes on the board are legal choices — and even a "friendly runes" wording (like Sona's) would still include your teammate's runes, because friendly covers objects controlled by you or your teammate (CR 491.8.e / 740.1.a). Battle Preparations has no errata changing that text.

One caution: control is still not shared, so you can't spend those runes yourself. Readying them only helps your partner — typically letting them pay for a Reaction spell or ability during your turn when you invite them in with priority (CR 491.8.a).