I see two issues here. One of them is that it is about 600 metres from Solomos Square bus terminus in Nicosia to the Ledra Street crossing point. This can comfortably be walked in ten minutes without luggage, but the question is how much luggage you will be carrying and you also need to realise that there are no public transport facilities immediately on the other side, either.
The other issue is that, as Lordo says, the inter-city buses from Larnaca take you to the Solomos Square terminus, but these buses leave from the centre of Larnaca, not the airport, and the airport shuttle buses to Nicosia take you to a terminus at Kyrenias Avenue on the fringes of the city centre, although there are regular city buses along the main road nearby that take you to Solomos Square.
So, if you want to do the journey entirely by public transport, you will either have to take a local bus from Larnaka airport to the town centre, and then take the intercity bus to Nicosia - Solomos Square, or take the airport shuttle to the Nicosia terminus and then get a Nicosia city bus to Solomos Square. Either way, you will need two buses, and then you will have a bit of a walk to the crossing point which may be difficult if you have a lot of luggage.
For your information:
Times of intercity buses between Larnaka and Nicosia:
http://www.intercity-buses.com/?wp=rout ... 05195E3DBFTimes of the local bus in Larnaka that takes you from the airport to Foinikoudes, where the intercity buses start from:
http://www.zinonasbuses.com/admin/uploa ... pdf277.pdf(Somebody has posted the times of the Larnaka Airport - Nicosia shuttle)