I have been running linkedin ads for a couple of years, and spent quite a lot in it. So you could say I have experience.
In general terms, when you target smallish audiences LinkedIn tends to get really expensive (moreso than it already is) and quite inefficient, so rather than targeting specific companies I’d recommend you target companies by industry, size or category (this is, for example, fortune 500 companies, or forbes’ most innovative companies, among others), and then narrow down your audience a bit. Depending on what kind of product or service you are advertising, I would recommend to narrow down by job function (you might want to avoid showing your ads to interns for example), or by department. But there are many categories you can use to effectively narrow down your audience.
In my experience LinkedIn can be a good channel for brand awareness , but it is not great to get people to convert. So (and this especially applies if you are running with a brand awareness objective) don’t expect to get conversions or even many website visits. So you will have to measure your success by looking at how many people in your audience your reached, at what cost, and maybe by running a demographics report. If you are using Campaign Manager you can add an impression tracking pixel to your ads and see how they influence other media.
Finally, something I usually do, is to create a highly targeted campaign with a traffic objective, trying to lure the people I’m interested in to a linkedin-exclusive landing page, and when I reach about 500-1000 visits I create a Lookalike on Facebook. Facebook does a great job recognizing the patterns those people have in common, and lets you target lots of people who are similar to your LinkedIn audiences at much lower costs (needless to say, it’s not so accurate as LinkedIn of course, but given how cheaper it is I think the waste is acceptable). But I should say that, in general terms, I don’t target by company, so this tactic may not work very well in your case.
I have quite a lot of experience with LinkedIn ads, so if you need help or have any questions let me know! Good luck with your campaign.