Who pays for checking for damages?

I think it would come down to whether there was negligence in bumping into something - assuming we're talking about a bump with one's body.

You aren't clear about the nature of the item, the setting in which the bump occurred, or why there was a bump, but generally speaking the onus is on the owner of sensitive equipment to keep it away from visitors.

A simple stumble and crash into furniture in a domestic setting is not negligence, without more (such as age-inappropriate horseplay, with middle-aged adults being held to a different standard than children).

It is of course negligence for say a commercial piano mover to drop the piano, however.

I don't think the fact that the item isn't found to be broken after inspection bears, provided the handling was rough enough that internals may have been damaged, and a reasonable owner would "lose confidence" in the existing condition and be put to the labour of confirming or re-setting it.

If there was negligence, there would also be the question of whether hiring a professional was a reasonable alternative to a "time-consuming" self-check, if the owner could have done this themselves.

No one is liable for "accidents"

Accidents that damage people or property are at the owner's or the person's risk. It is not enough that your acts or omissions caused another's loss; to be liable, your actions must, at the least, have been negligent.

To be negligent, you must have owed a duty of care (which we'll take as given) and failed to discharge that duty reasonably. Whether a "bump" is a failure to discharge the duty depends on the circumstances; however, since you have described the bump as "accidental", your neighbour would have to show that you had a positive duty to do more than just move naturally. However, if you were driving a car when you "accidentally bump", then that isn't an "accident" because, as a car driver, your duty is not to run into things.

If you are negligent, then you are liable for the reasonably foreseeable loss. Again, what is "reasonably foreseeable" depends on the circumstances. If you are blundering around a University science laboratory, then it is reasonably foreseeable that any equipment you jostle might need checking, however, this would be the exception rather than the norm.

You would be legally responsible to pay if you had agreed to pay: or if you are ordered to do so my the court. The former is more likely than the latter, and involves OT social analysis rather than legal analysis.

If you did indeed damage the object through your negligence, you could be held liable for the damages that you caused. Let's assume that you completely caused the damage and the neighbor has not contributed to the damage. He can sue you and the courts can tell you to compensate him for his reasonable costs. One course of action A would be that he sends the object to a repair place that charges $1,000 for a cure no matter what. Another course of action B for him would be for him to send it to a different shop that charges $100 for diagnosis, and $1000 for cure. Either way, he would have to do something that costs him money, then he takes you to court to get his money. Given your proven responsibility, the only variable is whether his choice of solution A vs B is reasonable, and yes, either is reasonable. If there were a third option C where he pays $1000 for diagnosis and $1000 for cure, that would not be reasonable in the face of cheaper alternatives.