I think this is really too broad to get a proper Answer here but I'll try to give you some guidance to help you proceed.
I'm a big proponent of not reinventing the wheel and when it comes to standing desks, having been flavour of the month in certain circles in recent years, there is a lot available to draw from.
You can start with a plan or build-a-long online or in a woodworking book. An alternative is to approximately copy an existing standing desk of simple design and build that you see for sale, by measuring key dimensions and taking lots of photos from every angle, and going from there.
...the only tools I have are a hammer, level, and a decent impact drill with a small bit set.
I'm afraid this is not enough to do basic woodwork1.
You really have to have some facility to cut wood and even if you could buy all your wood cut to size (which often means said materials come at a premium2) you'd have no ability to make any adjustments that might be needed, alter something to suit an emergent situation or just to change your mind about an aspect of the design that comes to you during the building process..... or later after you've used the desk for the while and you have 'some notes' on the design ^_^
In general for basic woodworking you need some of the following:
At least one saw. I can't be more specific as the type of saw, whether it's manual or powered is very much up to individual preference and budget and what you're intending to cut.
Some measuring/layout tools. The most critical is a square of some type, as almost every piece of anything needs to be checked for square or marked for square, often multiple times.
Some marking tools. This is the simplest requirement as you don't need anything more sophisticated than a ballpoint and/or sharp pencil and a sharp knife for adequate marking out. Any sharp knife will do really, but if you need to buy something a basic utility knife or box cutter is fine and will have the most onwards utility3.
A few sanding supplies. At the most basic a pack of sandpaper and a sanding block. Good abrasive paper is generally worth it but you'll find it's more expensive than you're expecting. You can make a sanding block from just a cutoff of softwood, but various styles of sanding block (some of them plastic with metal clips to hold strips of paper torn from full sheets) are available to buy if that takes your fancy.
Further for this project specifically you may need some finish for the wood/wood products you end up using. Even if rated for exterior use the materials may benefit from having a little finish applied, e.g. to improve the way the surface ages with exposure to weather4.
I can't really give a good estimate of what these would cost because of all the variables, but even buying as cheaply as possible this is going to blow your budget, and you having even bought any wood or board materials yet. So some adjustment will be needed there unless you can source materials for cheap, or for free — from furniture left on the curb on trash day, at yard sales, castoffs or damaged pieces from building sites. And of course from the ubiquitous source of free wood, pallets. See Recycling "pre-loved" lumber -- techniques? and What do I need to know when looking at pallets for wood? for a bit more on this.
1 Old training manuals and beginner woodworking guides give an idea of what you might need to build the simplest starter projects and this will often include two saws, at least one hand plane and some or all of the measuring, layout and marking tools I've gone on to list, as well as a drill or drills and a range of bits.
I'd forgotten about these previous Questions until now, some more on what's needed to get started in the following:
What is a good starter traditional hand-tool set
What is the best way to get started with limited funds and space?
2 Not because of a cutting charge (a few cuts are often free) but because the type of place that will offer this service is often more expensive to begin with. And this is for board materials only, few (if any) places will cut 1x or 2x material for buyers.
3 Rather than a dedicated woodworking marking knife, which work great but are really only for that.
4 Assuming the desk is left permanently outdoors. I would suggest making something that can be taken indoors, if necessary by being a knockdown (often shortened to KD) design so that it can quickly be taken apart and moved as major subassemblies, at least to take the working surface(s) inside.