A VPS is definitely overkill if your utilization rate is low. You might be able to swing a deal with a server operator over at LowEndTalk or use something like AWS EC2's spot instances or on-demand instances (depending on your exact requirements) or Microsoft Azure. You pay by the hour/minute for EC2 and Azure. I haven't spun up an EC2 instance because Microsoft provides my company with a bunch of free credits (obligatory plug for sponsor!) so I can't say how easy it is (it's straight forward to use, though -- I've used EC2 instances before). Having been accustomed to using VirtualBox's provisioning system, I found Azure's provisioning system to be a bit more complicated, but not overly difficult.
A standard Azure VM with 5GB of RAM would costs 28 cents/hour (billed by the minute). So that would set you back about $30/year for your 5 minutes of usage per day, plus a bit extra for the storage (I'm assuming you want persistent storage) and your initial setup. So maybe $40/year. This comes with a Windows license, should you want it.
I'm not familiar with all the gotchas for EC2, but on-demand pricing for 8 GB of RAM and two cores is 9 cents/hour if you don't need Windows (it's a few cents more per hour if you do). If you don't need much storage, I think this is much cheaper. And there's not that much bandwidth I expect you're going to consume in 5 minutes/day. So this probably works out to about $120/year if you need to run your job at 5 minutes per day (+ rounding up from Azure's per-hour billing) (or less than half of that using spot instance pricing, if you're flexible on timing).
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Lambda maxes out at 1.5GB of RAM. If you can reduce memory requirements it'd be perfect for this use case, and it would probably come within the free tier.
– Tim – 2017-08-13T22:27:41.820