Does the Svelsongur work on Udyr abilities when I spent his buff to deal 2 damage on a unit at a battlefield which results as a 4 damage instead of 2?

No — you'll still deal 2, not 4.

Svellsongur gives Udyr a second copy of his printed ability, so he now has two separate instances of "Spend my buff: … Deal 2…". But spending the buff is the cost of that ability (cost: effect, FAQ #5276, #5122). There's only one buff on Udyr (Rule 702.3 — a unit can only have one buff), and you can't spend the same buff to pay for two abilities (FAQ #5276). So you can only activate one instance per buff, and it deals 2 damage.

This is the same interaction as copied exhaust-cost abilities — you can't exhaust once to pay for both copies (FAQ #6561, #5076). The second copy only matters if Udyr gets another buff later, which you could then spend on the other instance.