I am not a native English Speaker, but I work with English speakers on a daily basis and they have always troubles pronouncing my surname, so they will often ask me to tell them how they can pronounce it. My surname is Sulce. I also tried a pronouncing service and they said "Sorry, we can't pronounce that." So, I would like to ask native speakers to help on defining the most correct way to pronounce it. What's the first pronunciation that comes to your mind for the word Sulce?
The lack of clear and consistent pronunciation rules in English for a word like this is why people ask you how you pronounce it. The correct pronunciation of names is very dependent on the cultural context where you are from, and the way you pronounce it is the correct way. When somebody asks you, say what you would say when introducing yourself (like "My name is ____ ____") and then they can hear the pronunciation. If you are yourself unsure, your parents and/or other family members with the same surname are likely the best folks to ask.
When I read your name, I have a few hypotheses about how it might be pronounced, but lack enough information about where you and/or your family are from.
My first guess is that it's like the Spanish "dulce" (links go to audio) but with an S instead of a D at the beginning. To some extent, the meaning of whatever word you tell people to use to remember your name pronunciation will become mentally associated with you; "dulce" means "sweet" which I think of as being a generally positive association but it depends a bit on your style and personality and what you're going for.
Dan Bron's guess that it's pronounced like "pulse" with an "s" instead of a "p" would probably be my second guess.
My third guess, [ˈsəl-sē] would give the final e more of an an i sound, like the first syllable from "sultan" followed by see. In that way it's kind of like "Sochi" with an "ul" sound (pull without p) instead of the o. This is similar to Peter Shor's guess, but a different explanation.
In light of the clarification "The way I pronounce my name in my native language is sʊltʃe. The last e is pronounced like 'e' in get." in comments:
Then a pragmatic textual way to instruct English speakers would be to write that the name "rhymes with PULL-chay" or "rhymes with pull-CHAY", depending on the stress.
The English "AY" may not sound to you especially similar to the "E" of your name, but I assure it's subjectively similar to most native English speakers, and it is as close as a reasonable majority of them will get to with simple instructions.