As far as NAT or inbound flows, not enough detail was provided here.
For host gateway redundancy a FHRP is typically used. And yes, VRRP is the open standards flavor you would probably be using.
OSPF is a routing protocol and through some engineering can provide different fail over mechanisms for your routed network, i.e. your gateway could have two ospf routes to decide which circuit will be used outbound.
Depending on how many inside VLANS(networks) you have my suggestion for you would be significantly different. Keep in mind, you have tons of options.
If you have tons of inside VLANS, it would suck to try to run VRRP on all of the VLANS individually(more config, management, and more keepalive traffic). What you could do here is plug both routers into a pair of switches to the south, use the switch stack as the L3 gateway for your hosts and use vrrp or OSPF(one or the other) to load balance(active/active)/failover(active standby) as you see fit(requiring only configuring FHRP on one vlan). The benefit here is obviously that your gateway never changes, only the preferred route outbound.
For example, if you had a pair of 4506E swicthes you would need FHRP on the side attached to your hosts, and maybe they would use OSPF upstream to prefer a certain route or load balance out both. A pair of 4500x would differ because they are stacked, they would not need FHRP, but could still use some form of dynamic routing (OSPF) to choose the best outbound path.. Just examples to show you how many options are truly available.
If you don't mind doing it/don't have lots of host VLANS, you could configure VRRP on both routers for each vlan you intend to provide redundancy for.
If we use VRRP, would it be a good way to assign the VRRP VIP address as default
gateway to the hosts? Correct.
For example, you would assign one router an IP address 10.0.0.2/24 and the other 10.0.0.3/24, they could then use VRRP to share the 10.0.0.1/24(which would be assigned as the default gateway for your hosts in that network) providing 'first hop redundancy'.
Keep in mind that FHRPs are active/standby(exception being GLBP). This means that you will likely waste some, if not 50% of your total bandwidth; assuming that you don't engineer it properly. Example, all even VLANS use one primary gateway and all odds use the other.
Load balancing will likely be the preferred option.