Is there posibility to re entry Japan?

In 2013 I went to Japan with visit visa.while I have my visit visa I changed it to designated visa.in 2016 I returned to my country because
at that moment my mom's situation was not good. when I return my home I already had designated visa.it was not expired. I spoke with designated activity department of immigration beurau in Tokyo and I have cancelled my application then return back to my country.
but now I want to go back Japan. Is there a posibility to get visa for me?

You said you 'cancel application', which I guess your designated visa/status is related to your 'cancel application?

So when you cancel application it also cancel/void your designated status?

Did you submit special re-entry permit at airport, when you travel back to your country?
http://www.accessj.com/2013/09/the-new-re-entry-permit-system.html

With that special re-entry permit you should be able to re-enter Japan within one year as long as your status is still valid.

Yes, you can apply for new visa but no one can say whether you will be given/granted visa.
You can try.

Would it be possible to reenter Japan as a tourist, but with valid designated activities (multiple entries) visa?

Why would you want to enter Japan on temporary/tourist visa when you have multiple-entry designated visa?
Do you want to hide something?
You may be asked to explained that by immigration officer, being suspicious.
Possibility of your designated visa may get revoke/cancel.

No one can give you the correct answer here. Go ask Japan embassy, or Japan immigration office.

No, nothing to hide here. I just want to use the JR Pass and I don't see the point in paying much more than any other tourist only because my visa is too long for my stay. I have enough documents to proof, that I will have finished my internship by then and already have a flight back. The question is, do they allow me to enter as tourist? Unfortunately I couldn't find any information on that anywhere and just to try it might be not worth it.