Using 'eye up' suggests a strong interest or desire for someone or something. After the presentation, he eyed up the award, hoping to win it.
Pro Tip 2/3
Slight Intrusion
'Eye up' can feel intrusive if used about a person, as it may imply undesired attention. She felt uncomfortable as he kept eyeing her up from across the room.
Pro Tip 3/3
Casual Observation
'Eye up' is casual and might not fit in formal contexts. Use in everyday, relaxed situations. At the flea market, I eyed up a few vintage watches.