Hang on there! I don't think you are going in the right direction at all. I think you may have misunderstood things a bit.
Plesk is a hosting control panel. It effectively takes over a server's web, email, DNS, well...almost everything. You use it to create website hosting and email accounts for one or multiple domains, either for your own use or to sell as a service to others.
You will typically find Plesk pre-installed when you rent a server, or it may be available as an extra-cost option, or there may be some other hosting control panel pre-installed. For example GoDaddy offers Plesk, or alternatively their own hosting control panel, or yet another hosting control panel called cpanel, and in all three cases there is the option to install various pre-packaged software applications at the touch of a button.
If you had Plesk, and used it to install an application on your VPS you would surely know about it. When you logged in it would say "Plesk" in various places that are too difficult to miss.
On the assumption that you do not already have Plesk, you most certainly do not want to install it just to deal with DKIM and SPF. And if you do have Plesk already installed and didn't notice or know about it then you definitely don't want to try upgrading it like this. It is not a tool or utility that you might add to an existing server currently doing an existing task.
It was not clear from your post exactly what you have at the moment... Maybe some more information would help figure out what you have?
I know you have a VPS. What do you do on this VPS? What is its purpose? How do you control it? You mentioned an "app" - what does that do? What is it called? How did you install it?
Based on no evidence at all (i.e. I'm totally guessing), I rather suspect that you may have some other control panel, maybe a custom one provided by the VPS provider, and used that to install an application on your VPS? Possibly?
Does this application you have installed connect in some way with Mandrill? How did you make the connection?
But ignoring all of the above......:
When it comes to SPF, all that's required is to create the required DNS records in the DNS server responsible for your domain name.
DKIM is a slightly different kettle of fish. With DKIM, a sort of cryptographic hash is generated *by the sending server* (mandrill in this case, I presume) and added to the email's headers. But in order for this to work, a specific DNS record must again be added in a DNS record for your domain.
This page:
http://help.mandrill.com/entries/22030056-how-do-i-add-dns-records-for-my-sending-domains
tells you exactly what needs to be added for both SPF and DKIM for use with Mandrill.
In both cases, the DNS records must be added on the DNS server that is responsible for your domain. This is the key to your problems.
I note that at the bottom of the page I referenced, where there's a GoDaddy section, the only information provided is how to add those DNS records in Plesk 9 or 10. And I think this is the source of your confusion. Those instructions only apply if you already have Plesk 9 or Plesk 10 installed AND that Plesk installation is responsible for the DNS records for your domain name.
If you don't have Plesk installed, you don't want to install it. It will not help. It will probably break your VPS which probably has some other control panel already doing everything. And even if you do have Plesk installed, I would suspect it is not responsible for the DNS records for your domain unless you made specific changes to the configuration of the domain at GoDaddy, which would involve creating nameserver host records then changing the nameservers associated with the domain to your newly-created host records).
Instead I would suspect all you probably need to do is login to your GoDaddy account and look for the DNS settings section for your domain name, and make the required changes there. This is on the assumption that GoDaddy's DNS servers are responsible for the DNS records for your domain, which is how things are by default when you buy a domain from them and don't go changing stuff afterwards.
Now all of this is based on guesswork and reading between the lines, so it is possible that none or little of what I've said is of any use or even correct. So I strongly suggest you contact the Provider of your VPS, explain what you have now, who provides it, and what you want to achieve, and get them to help you untangle this confusion and get to the right solution.
Again I will emphasise this--- most of what I've said is based on total guesswork, and I could be totally and utterly wrong about everything.