If I understand your question correctly - no you cannot upgrade the level of your weapon. You can add +s to it with weapons elixirs but it would be a huge waste at lower levels.
Much cheaper to purchase lvl appropriate weapons from stalls (not NPC)as you go. I wouldn't think enhancing a weapon under lv20 would be worth it unless it's specific sp farming equipment that you will use for a long time. Generally you just move through the lower levels so quickly! Personally I didn't enhance a weapon until I got my lv 32 SOS blade. (drop from an ong - love ongs
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Save your weapons elixirs for the high levels when you will most likely need a lot of them! Alternatively you can sell them for $220k- 250k each so you have money as you go along.
Good luck and safe leveling
. you cannot up the lvl, you can better get to lvl 13 and then buy a lvl 13 blade/shield from a stall 
