I'm fluent in Japanese, but I've just started learning Korean, it's fun finding connections. I actually found this forum because I wondered if Korean 다라 (tari, meaning leg) and Japanese 足りる (tariru, meaning to be enough, but it uses the kanji for leg) were related and found a post about that. The majority of similar words are Sinitic and there are quite a lot of them. Some words are pretty obvious, like 주의 and 注意　(both chuui) others are not similar enough to be obvious without context or a dictionary, like 확인 and 確認 (hwagin and kakunin). I'm curious if there are any common patterns besides the ones I've noticed.

っ and つ (small tsu and tsu) generally become ㄹ (l)
Examples: 특별/特別 (teukbyeol / tokubetsu) 벌표/発表 (beolpyo / happyou) 설치/設置 (seolchi / secchi)

く　(ku) generally stays a k sound, becoming ㄱ (k)　
Examples: 세탁洗濯 (setak / sentaku) 입력/入力 (ibryeok / nyuuryoku)

long o and long e おう and えい often becomes ㅇ (ng)
Examples: 중학교/中学校 (chunghakkyo / chuugakkou)　동부/東部 (dongbu /toubu)　설명/説明 (seolmyeong / setsumei) 선생/先生 (seonsaeng / sensei)

Let me know if you know any other general patterns that might be helpful to know. ^^

Things are easier if you know pre-modern readings of kanji.

法, read ほう hō, was originally はふ fafu; cf. Korean 법 /pəp/.
方, read ほう hō, was originally はう fau; cf. Korean 방 /paŋ/. Note that historical Japanese au reflects Middle Chinese -Aŋ, where /A/ is some open medial.
業, read ぎょう gyō, was originally げふ ŋgefu; the Korean reading 업 /əp/ reflects Middle Chinese initial ŋ- as zero, while Japanese retains it as [ŋg] (now pronounced [g] word-initially).