I have been wondering about that as well and here is what I could find.
I read about trips with several major container ship operators (CMA-CGM, Hanjin Shipping, MSC, NSB) plus some roros and Polish bulkers. I also found many agencies selling more-or-less the same trips (a few in Germany, one each in France, the US, Australia and the UK). In all cases, the prices were around EUR 80-110 per day, plus deviation insurance and embarkation fees (which are more-or-less fixed costs, so they weigh much more heavily on short journeys).
Those are the only companies I have heard of that verifiably take passengers on a regular basis and they all have pretty similar prices/conditions. Thus any trip across the Atlantic or Pacific, to or from Asia or Oceania or through the Suez canal will cost at least €2000-3000, sometimes much more and circumnavigations start at €10,000.
Consequently, it seems the price does not depend so much on the ship, the agency or the operator but mostly on the route/length of the voyage. At EUR 100 per day, ocean crossings or longer journeys will always cost a few thousands euros/dollars. Voyages to or from Australia and New Zealand also seem in particularly high demand.
The only other option seems to be shorter “partial” journeys, hopping on and off a ship doing a longer rotation. Nothing dirt cheap and certainly nothing cheaper than a commercial flight but still some opportunities to travel on a freighter for less than EUR 900/USD 1000. Some examples:
- Partial journeys on Grimaldi Lines' “Euromed” roro tours, for example Salerno (Italy) to Setubal (in Portugal) or Salerno to Limassol (Cyprus) or Ashdod (Israel). Salerno-Setubal is €285 per person in an internal cabin with a double bed or €315 in a cabin with three single beds (but I think you have to find your travel companions yourself and book the entire cabin together, unlike couchettes on trains). That's the absolute cheapest price I came across. Some agencies also list the Italy-Cyprus trip, usually with a small markup.
- Rotterdam-UK and return (no one-way journey) on a container ship, €345 per person through Kapitän Zylmann (price includes insurance and admin fees). Kapitän Zylmann offers another round trip, Portugal to Morocco and back on a small container ship, for €560. According to the website, it only calls at one port in Morocco where the whole cargo is unloaded so it stays longer than usual in that port, which should allow passengers to disembark and explore a bit before returning to Portugal.
- Rotterdam-Dublin-Rotterdam through Frachtschiffreisen Pfeiffer, starting at €490, two days at port in Dublin, apparently no one-way crossing available but it should be possible to go on land. The same agency has a number of one-week trips to the British Isles or across the Baltic also around €500-600.
- Fos-sur-Mer (near Marseille) to Malta, with a container ship from CMA-CGM. Listed by Mer & voyages as “Med 1287” for €570 (but that does not include insurance and admin fees). It might also be bookable directly from the ship's operator, possibly a bit cheaper. If I am not mistaken, the ships on that line continue to Suez and then Asia. CMA CGM operates some of the largest container ships in the world and they are much larger than the ships listed above so it might be a good way to experience what modern shipping is like.
- Cartagena to Guayaquil, over the Panama canal, is listed for €800 by Kapitän Zylmann. From the same agency, it's apparently possible to book Cartagena to Puerto Cabello in Venezuela for €610 including fees and insurance. In the same region, NSB sells journeys on container ships taking passengers and calling at Cartagena, Port-of-Spain, Kingston, Manaus, etc. They don't list many details on price or conditions but it might possible to create some interesting shorter journeys from those lines.
- Antwerpen-Istanbul with a container ship from MSC, also bookable through Pfeiffer. A bit more expensive at €810 plus fees and insurance (in total EUR 1100/USD 1250/AUD 1600), but still close to €1000 and a meaningful journey, crossing Gibraltar and the Mediterranean, 9 days (only double beds so here again travelling alone is more expensive). Slightly cheaper but following a similar idea, it's possible to go from Hamburg to Genoa for €785.
From what I can tell, some agencies are a little more reluctant to sell short trips than others but it seems that it's often possible to book a partial journey on a given freighter line. So you could also check the longer journeys on all the websites listed here and then approach an agency to enquire about a specific trip, even it's not listed explicitly.