As pointed out in other answers, wear and tear of work equipment is generally the employer's problem. In germany specifically, the law is even more in the employee's favor: In general, employees are only (fully) liable for gross negligence.
It is well-established by court precedent that for all activities that are part of an employee's duty, the employee is not liable for damages, even if the employee was negligent - unless the employee was significantly negligent (or acted with malice). The German Wikipedia article on Arbeitnehmerhaftung explains this in more detail. Even in case of significant negligence, liability is often limited.
Additionally, in your case it is not even clear you caused the damage (as the laptop is used, and may have been damaged by the previous user). So to make you liable, your employer would need to establish that
- you actually damaged the laptop, and
- you did so out of negligence (i.e. it was not just regular wear)
No employer is going to bother with that, they will just eat the cost.
To address your specific concerns:
I know that large companies have It departments for dealing with issues like this but I work for a startup and I'm in my notice period, and I'm concerned they'll try to make me pay for it [...]
Try not to worry about problems before they become relevant :-). If your company does not have a dedicated IT department, they must still be someone who is responsible for the equipment. Find out who that is, talk to them, and hear what they say.
I already declined an offer from my boss to buy the laptop so I think he wants to recoup the cost of the laptop.
While that is possible, it does seem rather far-fetched - a) a used laptop is not worth a lot, so there is not a lot to "recoup" anyway, and b) your boss will not be able to claim the full cost of the laptop if only one key is damaged.
Also, it is equally possible your boss just wanted to do you a favor by offering you a laptop for little money.
While it is good to try to anticipate problems, try to not assume the worst :-).