Is there a way to make GitoLite work through passwords rather than through public+private keys? For example, to make it in such way that the user can clone the repository only through his/her own gitolite password confirming his/her identity, not by generating a key pair for that.
No. The entire point of gitolite is that it uses keypairs to determine someone's identity, so that it can manage users, instead of having to manage them on the system.
You could bypass gitolite by giving some users shell access to the repository directories, and having them push directly via ssh (and authenticate with their method of choice, e.g. passwords) but this really would be bypassing it. Gitolite would no longer be able to do all of the fancy things it does, namely checking to see if a given user is authorized to perform a given operation.