Go even if not everyone else is going.
If it turns out I'm wrong, going once won't hurt your career, even if you don't get paid for it. You could even leverage a non-productive unpaid office cleaning exercise as an example of why you are in the future as careful about always getting paid as some feel people should be.
The people most integrated to the team will be going, unless they're otherwise unable to go. That likely includes the effective team leader, whether or not that's your manager. And it's team-building for everyone who does go.
We had an event like this at my first career job. This was, admittedly, in the US (Texas), and I was working for a contracting company, and this was at the contracting company office, rather than where I worked.
I expected it to be a team-building event. Two people showed up for it. I expected my manager to show up, at least, since he'd been the one to invite everybody. He wasn't there. The woman who was the other person to show up wasn't somebody I even recognized, but she somehow knew me, at least by name and contract.
She explained why three people weren't there. One of them was a sales manager, but the other two were just people I vaguely recognized as office staff. After waiting a bit longer, she said she guessed it would just be us, and we got to work.
She and the other three people she mentioned turned out to be the driving force of the office. She was the office manager, and one of the other people was the HR person I hadn't already met. Partially because I went there and gave it my best, and continued to be involved in things like that, I never needed to worry about having a contract, because the people who mattered knew me, and if I was an option for a contract, I was one of the first people they considered for it.
Also, while I wasn't interested in management positions (I was well aware I lacked the people skills to be able to do that), I was considered for them briefly several times, and most of the people I knew who rose quickly through the ranks at that company went to those events.
That particular company had a lot of optional events to go to that you could use to impress the people in charge, so there were other ways to make such impressions. But if you want to convince your office team that you care about the company and the work environment, there may not be a better way to convey that than going to this thing, because almost nobody likes to clean.
Furthermore, if you don't go, then you don't get to be the one to clean your workspace, which means you have no say in how your workspace is cleaned.
That having been said, I also strongly recommend that you talk with your boss about how the work for that will be compensated. That was a thing that I had failed to do, as I come from a workaholic family, and I was more concerned about making an impression about being a hard worker than I was for getting paid for every second.
But my manager called me after I turned in my next timesheet, confirmed that I had gone to the event, and reprimanded me for failing to fill out my timesheet correctly, and for having gone over on time for the week without approval. These things were considered important. It was certainly better for me to have gone even without having gotten the timesheet business right, but it would've been better if I'd have gotten that right, too.
If this office cleaning day is at the customer office rather than your employer's office, and your customer boss says it's unpaid time, double-check with your employer boss before going to this event.
Different companies are different, so rather than assume the company is a certain way, you should ask.
Note: Depending on who exactly is in your office, it may not be nearly as important career-wise as the event that I'd gone to turned out to have been for mine. But it's still team-building that's most likely with some of the most influential people in the office, and it's likely to get the notice of people that matter.
Of course, that's only helpful if you actually do a good job. At one of the later office cleaning events I went to, an individual performed in a way that inspired my comment about the importance of having a say in how your workspace is cleaned. That particular individual managed to influence himself out of a job.