Yes, you can validate the signature online and provided that it is Valid you can Add Verification Information to a signature (right-click on the signature and you'll get this command) and then save PDF. The saved PDF will have in it all info it needs for validation. You must validate at the signing time, you cannot validate at the current time and get this working.
I need to correct my issue;
I receive a pdf, that I need to verify signature on.
If I do this on an internet computer, it works fine,
but how to verify the signature on a document that is offloaded the internet?
Internet is not available from the system where signature-identification is needed.
from your answer, it seem like this can be obtained if the one signing the document adds info to his signature?
Anyone can add verification information, not only the signer.
Validate the signature on an Internet-connected computer at the signing time (default since Acrobat 9.1). The follow the steps to add verification information in my previous note. Save PDF (with a different name or in a different location so that you still have the original in case anything goes wrong). You can validate signature in the second PDF offline.