According to Wiktionary, שלמה (pronounced /ʃloˈmo/ in Modern Hebrew) is the Hebrew version of Solomon. The pronunciation seems to follow reasonably well from the spelling, and as far as I can tell, it was spelled the same way in Biblical Hebrew.
The descendants look quite a bit different from the Hebrew pronunciation:
Ancient Greek: Σολομών (Solomṓn), Σαλομών (Salomṓn), Σαλωμών (Salōmṓn), Σολομῶν (Solomôn), Σολωμών (Solōmṓn) → Gothic: 𐍃𐌰𐌿𐌻𐌰𐌿𐌼𐍉𐌽 (saulaumōn) Greek: Σολομών (Solomón) → Latin: Solomon → English: Solomon Classical Syriac: ܫܠܝܡܘܢ (šlemūn) Arabic: سليمان (Sulaymān) → English: Suleiman → Turkish: Süleyman
It seems reasonable to me that the /ʃ/ shifted to a /s/, and most of the languages added a vowel between the /s/ and the /l/, and that Syriac and Arabic changed the vowels.
However, I don't understand where the /n/ at the end came from. I'm not aware of any other Biblical names that ended in an open vowel in Hebrew and shifted toward having an n.