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Approximately 2 weeks ago, I booked and paid for a flight that was scheduled for April on board Adria Airways, a Star Alliance company. However, I found out today from the media that the company is discontinuing service connecting to the destination airport starting from March. I haven't been notified about that yet, but the media report is reliable as the flight has already been removed from the airline website. Instead only connecting flights with other Star Alliance members are shown.

The departing airport is located in a Schengen Area and EU-member country, whereas the destination is outside of both EU and Schengen.

What are my rights? From what I read online, I am only entitled to compensation when the flight is cancelled not more than 14 days prior the flight.

There is another company (not part of Star Alliance), that flights directly between the two airports. Is there any option to demand my ticket be changed to their service without extra charges?

Schedules and cancellations are governed under Article X of the Adria Airways General Conditions of Carriage, the contract you agreed to when you purchased a ticket with them. This, in turn, is subject to national law and to Regulation (EC) 261/2004 since the flight originates in an EU country.

As is the case with most airlines, one of two things will happen when a flight is discontinued in the middle of the schedule, which from the passenger perspective is treated as a cancellation (emphasis added inline):

Except as otherwise provided by the Warsaw Convention or the Montreal Convention, if we cancel a flight or fail to stop at your destination or stopover destination we shall, at your option, either:

a. Reroute you to your final destination shown in your ticket, under comparable transport conditions, at the earliest opportunity

b. If neither of the above alternatives is acceptable to you, we will make a refund in accordance with the provisions of Article
XI/2. If you hold a confirmed reservation and ticket and you are not
informed about the cancellation, you might have the right to
compensation. You are not entitled to any compensation if we can
prove that if the cancellation has been caused by extraordinary
circumstances which could not have been avoided even if all
reasonable measures had been taken. These remedies are the sole and
exclusive remedies available to you and we will have no further
liability to you except as may be provided by the Warsaw Convention
or the Montreal Convention. If you are travelling free of charge (or
at a special reduced fare not available to the public) you might not
have the same rights as other passengers (exception is your travel
under frequent flyer programme). If you are travelling on a charter
flight your rights may differ from those on scheduled flight in
respect of ticket refund.

Adria is still advertising LJU-SVO flights on their English language website, so this may be a very recent announcement that has not yet percolated through all of their systems. Do note, however, under X/1,

b. Before we accept your booking, our authorised agents or we will tell you the scheduled departure time of your flight and it will be shown on your ticket. We may need to change the scheduled departure time of your flight after your ticket has been issued. You will provide us or our authorized agents with contact information and we will inform you of any such changes.

The airline is obligated to contact you to inform you of the schedule change, but depending on how you booked the ticket, there is a chance they have not received all your contact information, or they may only have informed your travel agent or booking website. Do not wait for them. You should contact Adria directly to discuss what to do.

If there is an alternative flight on a partner airline that works for you, you can certainly request that you be booked on it instead, for no additional charge, though you can't demand it. For example, that alternative flight might already be sold out in your class of service. Further, if Adria offers you a reasonable connecting flight on an alliance partner, you can't demand that they book you on a non-stop with a non-partner, though you can request it.

If they cannot find you an acceptable alternative, you are entitled to get the money you already paid (fare as well as taxes and fees) refunded. After all, they have not fulfilled their part of the contract. Because this cancellation occurred more than 14 days before the flight, however, you are not entitled to any additional compensation either under the airline contract or under EU directive, beyond the money you have already paid.