Reasonably priced day-trips in Svalbard

The 100-500 Eur range is right. At the expensive end there is 2-6 persons per guide and rented individual equipment. For NOK 800 you can get a day cruise on the fjord. You won't find anything cheaper, other than renting a gun and making your own trip.

Is there snow in June? Possible to ski?

There is probably some snow left in (early?) June. There is probably so little snow that snowmobiles are banned. You might find places to ski, you might have to carry skis from Longyearbyen and up to the heights. The snow you find might be heavy and wet. Skiing in such conditions is for the hell of it, and not because of good conditions.

It's impossible to predict the weather. But my guess is either snow with patches of bare ground or bare ground with patches of snow in mid june. Beware of large areas of wet (muddy!) ground.

While it might be more difficult than usual to get around during the snow melt, it is truly a unique time of the year. The terrain changes from day till day and new life suddenly appears.