Can I use my old usb cable on iPhone 13? Will my phone get damaged over time?

You can use the old cable. It won't damage the phone. Apple encourages you to use it. See the following from https://www.apple.com/shop/buy-iphone/iphone-13

As part of our efforts to reach our environmental goals, iPhone 13 and iPhone 13 mini do not include a power adapter or EarPods. Included in the box is a USB‑C to Lightning Cable that supports fast charging and is compatible with USB‑C power adapters and computer ports.

We encourage you to re‑use your current USB‑A to Lightning cables, power adapters, and headphones, which are compatible with these iPhone models. But if you need any new Apple power adapters or headphones, they are available for purchase.

Good one. I suspect the "environment" being supported by the new less equipment policy is actually called "Earnings per Share." This explanation assumes anyone buying a new iPhone is trading up from the previous model. No first time iPhone users or upgrading from, say iPhone 7 Plus, are expected to buy new iPhones. I, for example, went from a 4 to a 7, and now a 13, and I have tons of power adapters from Apple and others. Most are 1 amp, or 5 watts. A couple have 2.4 amp jacks, but they are USB-A.

So like apparently many iPhone 13 buyers, I have to also buy at the least a power adapter, and if I want to use the wireless charging, I have to buy a charger and a power adapter. So Apple wins in the EPS arena but drops a couple notches in the goodwill space. So...I bought an adapter and charger from Apple. The charger shipped, but the power adapter did not, although my credit card was charged for both. Another lesson. I could have bought both from Amazon and had same day delivery, and Amazon doesn't charge your credit card until it ships. That's two negative goodwill hits for Apple.