A good outboard should have no trouble running all day with no problems. You will want to make sure you have fresh fuel and I would run it for an hour at least around Olympia to get a feel for the boat. Would the seller agree to go with you, it would really help going through the locks if you had some help. Look up the currents for the Narrows, you don't want to be fighting a big flood, and riding an ebb could cut the trip time in half. The locks are pretty easy but if you haven't done it before you should look up the website and get familiar with the procedures. You should have fenders for both sides and at least two 50' lines. Listen to the lockmasters, they'll tell you what to do. Congrats on the new boat
I was just looking at Google Earth and it's about 60+NM to the locks from Oly. You aren't going to make that in one day at this time of year. Better find somewhere to stay near Tacoma. We used to keep our boat at Newport Shores in the lake, it is a three hour trip to/from the locks from there, you might make it faster because you won't have to stop for any bridges except Fremont.