I think you have at least two options:
- Get Apple Remote Desktop to run macs from another mac (in this case you just run server.app on the mini and control the mini from the laptop using ARD).
- Upgrade to mac mini to the same level as your laptop, ie. 10.9 and run server app from laptop.
I am doing #1 above with no problems at all. All my machines in the SOHO are running mavs but the two mac mini servers that I have (10.7.5). I won't upgrade them, as they work and I am not sure what breaks if you do upgrade. Best of luck!
It is standard in the software industry to have to pay for software upgrades. While Apple have chosen to make the Mavericks (10.9) upgrade free, the matching Server.app upgrade is not free.
You are not forced to upgrade to Mavericks, you can if you wish continue to use Lion (10.7) and its matching Server.app version. What you do need to do is use matching versions.
It should be noted that the 'full' cost of Server.app at only $19 is ludicrously cheap - far cheaper than the upgrade cost of most software.
In terms of using Server.app on your Mavericks laptop to administer your Lion server and the fact you can't run the Lion server version on your laptop, it is possible to use the Mavericks version of Server.app on your laptop to administer a Lion or Mountain Lion version of Server.app. Therefore you could buy the new Server.app for your laptop, leave your server still on Lion and use the new version to administer your server.
Or as coocooforcocoapuffs said you can use Screen Sharing or ARD to remote control your server and use Server.app on your server for administration.