Song lyrics are tricky to understand most of the time, because the songwriters don't have to use traditional word meanings or grammar to form their phrases if they don't want to. I love this song though! I can give you my interpretation of "take on me," but that doesn't mean it's how the songwriter meant it. My interpretation of that phrase is that it is similar to "take me on," which is also used in the chorus. To "take someone on" is to accept them and build a relationship with them, in this song. The songwriter is trying to convince someone to accept the challenge of falling in love with him, even though he will be "gone in a day or two." "Take me on" can also mean to fight someone (physically), but of course that's not the way the songwriter means it!