25 year old man, looking for accommodation in the Old Town of Vilnius

I'm a 25 year old man finish a study programme in project management in Oslo, Norway. I'm attending the Summer School in Vilnius in July, and are staying in the city until the 20th of December in conjuction with a student trainee program in international affairs. Although I plan to leave before Christmas, I'm willing to pay rent until the 1st of January, to make the rent period a full six months. I'm looking to rent an apartment within the 500-700 euro range, but can consider an apartment around 800 euros if it's a good fit.