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How does transaction fees work for a rollup? I mean, who pays for the L1 tx when the rollup syncs? Or when a rollup sends outbox messages to L1?

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There are two different mechanisms here:

The first, is that individual rollup nodes sync with L1 by posting commitments, participating in any refutation games, and cementing commitments. The L1 fees in this instance are paid for by the rollup operator - ie the person running the rollup node in question. These fees are paid by each rollup operator individually.

The second, is outbox message execution. Once a commitment has been cemented, any outbox messages on or before that commitment are now executable on L1 by anybody. The L1 fees are paid once, by the person executing that message.

The person executing the outbox message is likely to be someone with some vested interested in doing so - for example the message contains an L2->L1 withdrawal for a portion of their assets.

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