There is not a lot of information about the "cup of salvation" mentioned in this Psalm. However, here are some thoughts on the cup of salvation.
Here is a quote and a link regarding the Psalm but it is not that helpful regarding your precise question.
"Psalm 116 is a psalm that praises the LORD for answered prayer and promises lifelong praise for it. The psalmist can look forward to a long, tranquil life because the LORD proved Himself gracious to him. The psalmist had not lost faith in the great time of trouble, and so can now praise the LORD and edify others. In a word, then, the psalm is a praise for deliverance from imminent danger of death.
(from: The Hallel Psalms)"
The idea of "The cup" seems to have a particular reference to judgement: Judgement to destruction or judgement to salvation. To those in Christ, the celebration of the "passover" is duplicated in partaking of the true "cup of salvation" when we remember the shed blood of our Lord in the "communion" service week by week.
Here are some references to the cup - It seems to convey the fullness of the blessing or, conversely, the cursing by God of those who partake of the cup. The contents of the cup lead to intoxicating joy or destruction. Is it the same cup but the only difference being the state of the drinker? (like the stone in Matt 21:44 - same stone but different effects depending on the individuals involved)
We can have fellowship with God either by drinking deeply of the glorious cup of salvation or of the cup of His wrath when His fury is poured out upon a world lost in sin and degradation. One would prefer to be handed the cup of salvation rather than the cup of His wrath at the judgement day.
I hope you have found this helpful.
God bless and keep you in His way,