This is possible to get in the first 10 days. You can stack this with the Gone Green achievement for not eating meat for 10 days as well as the achievements for surviving 10 days, crafting a loom, creating a farm plot, and island hopping. If my attempt is anything to go by this will be a challenge.
Complete the tutorial but do NOT eat the cooked meat. Now it's time to get started. The life raft storage has 3 days of rations.
During those 3 days you need to be collecting coconuts on your starter island. Throw them on the ground next to your shelter. Collect sticks, fibrous leaves and stones. Keep everything in the pile. You can skin and drink/eat coconuts as needed. Do NOT consume more than 3 in a day or you will get diarrhea and become dehydrated and slowly die. You can survive if you get it but it makes the first 10 days harder.
When you can craft a crude axe. Find the recipe and click RS to pin it. Look at your pile of items on the ground and when the pinned recipe is full craft the axe. Now when the trees are out of coconuts start cutting them down into sticks.
Craft a crude hoe. Open your crafting panel and use RB/LB to find the farming plot. Craft one. On the starter island should be a potatoe. Plant the potatoe. It will take a few days to grow and will be watered after rain storms. You can eat one if needed when it grows.
Craft a loom, this is how you make cloth. You don't interact with the loom it adds the cloth rexipe to your craft page.
Now hopefully you have your crafting high enough to craft a larger raft. Open your craft page and use RB/LB to find the vehicles page. Craft 2 raft bases at 14 sticks each. Use RT/LT to change page and craft 2 raft floors, then craft a sail and rudder.
Place the bases down on the ground, snap them together then snap the floors down ontop followed by the sail and rudder. You can make your raft bigger if you want for more stability but I was able to do this with a 2 base raft. Using the life raft is also an option but it takes longer to paddle to the other islands.
You may be running out of drinkable coconut by now so the next goal is to craft the water still. You will need to craft 2 coconut flasks. One will be used to craft the still and the other will be your drinking container.
Save at your shelter before you leave your island. If your raft flips over quit and reload before you are eaten by a shark.
Sail to another island and collect more rocks, sticks, aloe vera for your SPF, and coconuts. Eat/drink if you need. More importantly check all around the beaches for blue squares on the ground. These are tarps and you need one to craft the water still. Once you have everything you need return to your starter island and craft the water still. Place palm fronds under it to keep the water collection going and use your extra coconut flask to hold 1 sip of drinkable water. Now you only need to worry about food, which means only eating coconuts so you have a smaller chance of pooing yourself to death.
By the time I crafted my still I was on day 9, dying of thirst with 1 life pip left due to jumping from a tree. Health only regenerates when your food and water needs are met and living off only coconuts and potates they usually are not.
Both achievements should pop at day 10