An inexpensive TIG box is fine to learn on, but it's nearly impossible to teach yourself to weld properly, let alone TIG weld. And TIG welding on round tubing that's 0.063 thin, is a major feat.
Now I'm not saying you can't do it, but it's not likely to look good, or be strong.
As for that unit, it only offers Lift Arc and no High Frequency start. That means you must first touch/scratch the surface to initiate the arc. Lots of machines and welders use this method, especially pipe welders when working with stainless. But it is harder for a new welder as you get a lot of contamination on the tungsten and stuck tungstens.
If you want an entry level machine that's easy on the wallet, but has some headroom, buy an Everlasting or Alpha; they're good "off brand" machines with quality features and decent performance.
Cup size is normally anywhere from a 5 (small tight spaces) to an 8 (more open area, walking the cup around tubing). Heck I even use a 12 for stainless steel for the added coverage. Your torch is probably a 17 style, air cooled, rated for 150 amps (WAY MORE than you'll use on the bike frame). Rule of thumb is 1 amp per 0.01 of thickness. So your hoop etc would be welded around 55-65 amps, using the pedal to soften the heat as you go.
Buying a bottle is expensive. Better to open an account a local welding gas shop, pay the annual rental on the bottle, then simply swap the bottle when empty (you pay a gas fee only). An 80cf bottle is good. You'll pay around $50 annually and about $35-45 for a refill. 100% Argon only for DC TIG!
Okay, here's the interesting secret: that machine (and all others like it that are DC only) say not rated for Aluminum. That's because using 100% Argon requires AC current for TIG. However, you can TIG weld aluminum with DC, even Lift Arc, but you MUST use 100% Helium as your gas. So, the annual bottle rental, you swap the Argon tank for Helium, pay the gas difference, and there you go, welding Aluminum too with a DC machine. Bobs your uncle!
On some of the Everlasting and Alpha boxes, you'll also have "pulse TIG" which is very handy for tubing and thin metal. It fluctuates the amps on and off at a tempo and percentage of amps to prevent overheating and blowout. Vey useful when you get rolling.
Other things you'll need: a quality auto darkening hood ($200-$300), gloves, jacket, grinder, flap disks, a silicon carbide wheel on your bench grinder or Dremel to sharpen your tungsten, and about a gazillion other little doodads to roll along. Oh yeah, and filler wire, practice metal, and... and... and...
Ain't welding fun!
Seriously, it's an absolute blast and once you learn, you will never look at anything again without saying, "I can make that for myself!" Spend the money and time and get one and have at it I say!