I make a quick search yesterday and you could make it by less than 400 Euros (if you depart the first day from Kaliningrad) although, perhaps, today prices will have got more expensive (and they can be even more expensive tomorrow..., you get the idea)...
The best way would be to look for one-way fares for each leg what means you have to make a separate search for each leg.
Using the multi-city option doesn't bring a cheaper fare in this case.
I've just checked out that multi city journey planner but it seems that the onward journey must be to a city within the country you originally left from
No, you can look for whatever multi-city routes you want (at least in Kayak and Momondo that were the webs I quoted you although you can make it in almost any web that allow searches for multi-city fares; e.g.: in Expedia too). Sometimes it shows you an airport by default but you can clear it and type your desired airport. However, as I told you above, it doesn't seem you can get the cheapest fare using multi-city fare this time (although you can give it a try). Therefore I'd suggest you to:
Look for a one-way non-stop flight Kaliningrad-Moscow on the first date. See what's the cheapest option (it'll depend on the date; i.e. if he prefers to fly May 26th or May 25th, as not all airlines fly that route daily). On May 25th, S7 airlines make that route and that's a good airline. Once you know what's the cheapest airline, go to their web and look there for the fare they are quoting and, if similar -or cheaper- to the quote you got from the search engine, book it. Kaliningrad is easy to reach by bus from Gdansk, isn't it? He could make it in a single day without too much hassle. You can also look for one-way options out of Gdansk and decide by yourself if the price difference worths the hassle of getting to Kaliningrad.
For the second leg, look also for a one-way ticket. He'll probably have to change planes but there are some options with good transit times. Look for the cheapest one (and skip those with awful transit times and overnight stops) go to the web of the relevant airline to see what fare are they quoting and, as above, if it's similar or cheaper than the quote you got through the search engine, book it in the web of the airline.
The third leg could be a good idea to take a look at Skyscanner firstly as it seems the best options are booking two separately one-way fares with a low cost airline (BTW, don't forget to verify the possible extra fees he would need to pay in order to know the real cost of the low cost flights). One of the options have a transit time between flights that I consider short for that type of tickets (remember I've you loose your second flight due to a late arrival of your first flight, you'll have to look and pay for alternatives on the spot and it won't be cheap) but there's another one with an IMO long enough transit time between flights. There are also some options with non-budget airlines that would guarantee you the connection if you miss your second flight due to a late arrival of the first one but they are more expensive... Skyscanner will show you all them. Don't forget to verify you are seeing all the options (specially, that you are seeing the non-protected options) though. If you finally choose an option including two low cost flights, don't forget you would have to book each leg separately in the web of the relevant airline so it's important you don't overlook in Skyscanner the airport where you would need to change planes with that option as you would need to book a flight from Bucharest to that airport and a different flight (and ticket) for the flight from that airport to Gdansk.
