Looking to put gutters here. 
This is how others next door are done, as well as how the ones I'm replacing were done previously. Basically a downspout connected between two gutter's via a hole cut in them.
Is this actually the correct way to do this, or is there a better way that doesn't seem so janky and leak prone.
Honestly gutters are something I pay for. Most good companies have machines that bend and shape them on site so you can make them continuous. That prevents leaks and ingress.
They also will have it done in a day, doing it my self would take much longer.
This chunk of roof is so short that there'd be no difference between hiring a company for continuous gutters and DIYing a segment of gutters.