3.7V Lithium ion battery supply and Arduino Uno
As noted by others, there is no simple way to boost a DC voltage, without an IC.

First, a short note on how the Arduino works. The Uno can use a Vin barrel jack that goes to a NCP1117 5v regulator, that needs at least 7v In for proper regulation. It can also use the VIN header pin for the same regulator. Or it can be powered via it's USB connector, where the USB 5V is almost directly connected, through a fuse and a switch (so that you don't break your arduino or computer, if usb and the Vin jack is used at the same time).

That said, you can simply use a Regulated 5v through the USB jack, instead of needing 7V, which would waste 2V * Current in wasted heat.

So you have a 3.7v Lipo battery. You will need a Boost Regulator to bring that up to a regulated 5V. You can get individual ICs, and the passive parts they need (caps, inductor, etc), or you can get ready made parts. 5V boost Regulators for batteries, with USB out, lockout protection (to prevent the LIPO from being drained so much it dies), and constant regulation are common to find. You can even get Arduino LIPO shields, or Lipo Charging and regulating boards. These can charge your Lipo from a usb port (and/or solar panels). And frankly, at about 10 dollars, is probably easier than designing and getting pcbs made. Boost regulator modules are cheaper, 4~6 bucks, and since you already have a lipo charger, it might be a better fit.